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PROFESSOR: Today I
want to get started

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00:00:25,330 --> 00:00:30,340
by correcting a mistake
that I made last time.

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00:00:30,340 --> 00:00:37,410
And this was
mistaken terminology.

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00:00:37,410 --> 00:00:41,290
I said that what
we were computing,

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00:00:41,290 --> 00:00:44,480
when we computed in
this candy bar example,

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00:00:44,480 --> 00:00:47,510
was energy and not heat.

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But it's both.

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They're the same thing.

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00:00:51,010 --> 00:00:55,320
And in fact, energy,
heat and work

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00:00:55,320 --> 00:00:59,470
are all the same
thing in physics.

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00:00:59,470 --> 00:01:03,895
I was foolishly
considering the much more

19
00:01:03,895 --> 00:01:08,350
- what am I trying to say -
the intuitive feeling of heat

20
00:01:08,350 --> 00:01:10,720
as just being the
same as temperature.

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00:01:10,720 --> 00:01:17,890
But in physics, usually
heat is measured in calories

22
00:01:17,890 --> 00:01:20,480
and energy can be
in lots of things.

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00:01:20,480 --> 00:01:26,610
Maybe kilowatt-hours or ergs,
these are various of the units.

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00:01:26,610 --> 00:01:31,840
And work would be in
things like foot-pounds.

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00:01:31,840 --> 00:01:35,500
That is, lifting some
weight some distance.

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00:01:35,500 --> 00:01:39,240
And the amount of force
you have to apply.

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00:01:39,240 --> 00:01:42,070
And these all have
conversions between them.

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00:01:42,070 --> 00:01:45,880
They're all the same
quantity, in different units.

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00:01:45,880 --> 00:01:54,030
OK, so these are
the same quantity.

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00:01:54,030 --> 00:02:00,850
Different units.

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00:02:00,850 --> 00:02:08,850
So that's about as much
physics as we'll do for today.

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00:02:08,850 --> 00:02:11,290
And sorry about that.

33
00:02:11,290 --> 00:02:18,800
Now, the example that I was
starting to discuss last time

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00:02:18,800 --> 00:02:21,980
and that I'm going
to carry out today

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00:02:21,980 --> 00:02:32,390
was this dartboard example.

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00:02:32,390 --> 00:02:40,160
We have a dartboard, which
is some kind of target.

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00:02:40,160 --> 00:02:46,380
And we have a person,
your little brother,

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00:02:46,380 --> 00:02:48,540
who's standing over there.

39
00:02:48,540 --> 00:02:50,680
And somebody is throwing darts.

40
00:02:50,680 --> 00:02:55,790
And the question is, how
likely is he to be hit.

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00:02:55,790 --> 00:02:59,950
So I want to describe
to you how we're

42
00:02:59,950 --> 00:03:02,550
going to make this problem
into a math problem.

43
00:03:02,550 --> 00:03:03,050
Yep.

44
00:03:03,050 --> 00:03:03,904
STUDENT: [INAUDIBLE]

45
00:03:03,904 --> 00:03:06,070
PROFESSOR: What topic is
this that we're going over.

46
00:03:06,070 --> 00:03:08,250
We're going over an example.

47
00:03:08,250 --> 00:03:11,650
Which is a dartboard example.

48
00:03:11,650 --> 00:03:16,090
And it has to do
with probability.

49
00:03:16,090 --> 00:03:17,370
OK.

50
00:03:17,370 --> 00:03:38,260
So what is the probability that
this guy, your little brother,

51
00:03:38,260 --> 00:03:47,700
gets hit by a dart.

52
00:03:47,700 --> 00:03:52,020
Now, we have to put some
assumptions into this problem

53
00:03:52,020 --> 00:03:53,630
in order to make
it a math problem.

54
00:03:53,630 --> 00:03:55,940
And I'm really going
to try to make them

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00:03:55,940 --> 00:03:58,630
pretty realistic assumptions.

56
00:03:58,630 --> 00:04:09,580
So the first assumption is
that the number of hits is

57
00:04:09,580 --> 00:04:21,520
proportional to some
constant times e^(-r^2).

58
00:04:24,430 --> 00:04:27,910
So that's actually a kind
of a normal distribution.

59
00:04:27,910 --> 00:04:30,560
That's the bell curve.

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00:04:30,560 --> 00:04:34,250
But as a function of the radius.

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00:04:34,250 --> 00:04:37,060
So this is the
assumption that I'm

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00:04:37,060 --> 00:04:39,990
going to make in this problem.

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00:04:39,990 --> 00:04:44,970
And a problem in
probability is a problem

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of the ratio of the
part to the whole.

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00:04:48,540 --> 00:04:53,580
So the part is where this
little guy is standing.

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00:04:53,580 --> 00:04:56,670
And the whole is all
the possible places

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where the dart might hit.

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00:04:58,232 --> 00:04:59,690
Which is maybe the
whole blackboard

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00:04:59,690 --> 00:05:02,670
or extending beyond,
depending on how good an aim

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00:05:02,670 --> 00:05:07,780
you think that this
older child has.

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00:05:07,780 --> 00:05:11,730
So, if you like, the part
is-- We'll start simply.

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00:05:11,730 --> 00:05:14,290
I mean, this doesn't
sweep all the way around.

73
00:05:14,290 --> 00:05:17,170
But we're going to talk
about some section.

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00:05:17,170 --> 00:05:18,930
Like this.

75
00:05:18,930 --> 00:05:22,400
Where this is some radius
r_1, and this other radius

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00:05:22,400 --> 00:05:24,460
is some longer radius, r_2.

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00:05:24,460 --> 00:05:27,830
And the part that we'll
first keep track of

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is everything around there.

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00:05:29,470 --> 00:05:31,470
That's not very well
centered, but it's supposed

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00:05:31,470 --> 00:05:35,060
to be two concentric circles.

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00:05:35,060 --> 00:05:38,840
Maybe I should fix that a bit
so that it's a little bit easier

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00:05:38,840 --> 00:05:39,784
to read here.

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00:05:39,784 --> 00:05:41,260
So here we go.

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00:05:41,260 --> 00:05:46,700
So here I have radius r_1,
and here I have radius r_2.

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00:05:46,700 --> 00:05:49,610
And then the region in
between is what we're

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00:05:49,610 --> 00:05:56,050
going to try to keep track of.

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00:05:56,050 --> 00:05:59,170
So I claim that we'll
be able to get--

88
00:05:59,170 --> 00:06:04,100
So this is what I'm calling
the part, to start out with.

89
00:06:04,100 --> 00:06:05,570
And then we'll take
a section of it

90
00:06:05,570 --> 00:06:12,220
to get the place where
the person is standing.

91
00:06:12,220 --> 00:06:22,157
Now, I want to take a
side view of e^(-r^2).

92
00:06:22,157 --> 00:06:23,740
The function that
we're talking about.

93
00:06:23,740 --> 00:06:28,650
Again, that's the bell curve.

94
00:06:28,650 --> 00:06:31,900
And it sort of looks like this.

95
00:06:31,900 --> 00:06:36,090
This is the top value,
this is now the r-axis.

96
00:06:36,090 --> 00:06:39,490
And this is up, or
at least-- So you

97
00:06:39,490 --> 00:06:41,540
should think of this
in terms of the fact

98
00:06:41,540 --> 00:06:46,900
that the horizontal here is
the plane of the dartboard.

99
00:06:46,900 --> 00:06:49,390
And the vertical is
measuring how likely

100
00:06:49,390 --> 00:06:52,460
it is that there will
be darts piling up here.

101
00:06:52,460 --> 00:06:55,780
So if they were
balls tumbling down

102
00:06:55,780 --> 00:06:57,690
or something else
falling on, many of them

103
00:06:57,690 --> 00:06:58,530
would pile up here.

104
00:06:58,530 --> 00:07:02,890
Many fewer of them would
be piling up farther away.

105
00:07:02,890 --> 00:07:06,430
And the chunk that
we're keeping track of

106
00:07:06,430 --> 00:07:09,200
is the chunk
between r_1 and r_2.

107
00:07:09,200 --> 00:07:11,590
That's the corresponding
region here.

108
00:07:11,590 --> 00:07:14,180
And in order to
calculate this part,

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00:07:14,180 --> 00:07:20,500
we have to calculate this
volume of revolution.

110
00:07:20,500 --> 00:07:21,350
Sweeping around.

111
00:07:21,350 --> 00:07:24,160
Because really, in
disguise, this is a ring.

112
00:07:24,160 --> 00:07:26,480
This is a side view,
it's really a ring.

113
00:07:26,480 --> 00:07:33,100
Because we're rotating
it around this axis here.

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00:07:33,100 --> 00:07:34,990
So we're trying
to figure out what

115
00:07:34,990 --> 00:07:36,800
the total volume
of that ring is.

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00:07:36,800 --> 00:07:39,930
And that's going to be our
weighting, our likelihood

117
00:07:39,930 --> 00:07:41,390
for whether the
hits are occurring

118
00:07:41,390 --> 00:07:51,240
in this section, or in
this ring, versus the rest.

119
00:07:51,240 --> 00:07:53,750
To set this up, I
remind you we're going

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00:07:53,750 --> 00:07:57,570
to use the method of shells.

121
00:07:57,570 --> 00:08:03,030
That's really the only one
that's going to work here.

122
00:08:03,030 --> 00:08:07,990
And we want to integrate
between r_1 and r_2.

123
00:08:07,990 --> 00:08:12,280
And the range here is that--
Because this is, if you like,

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00:08:12,280 --> 00:08:13,470
a solid of revolution.

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00:08:13,470 --> 00:08:17,250
So the variable r is the same
as what we used to call x.

126
00:08:17,250 --> 00:08:18,924
And it's ranging
between r_1 and r_2,

127
00:08:18,924 --> 00:08:20,340
and then we're
sweeping it around.

128
00:08:20,340 --> 00:08:25,920
And the circumference
of a little piece,

129
00:08:25,920 --> 00:08:35,040
so at a fixed distance r
here, the circumference

130
00:08:35,040 --> 00:08:37,950
is going to be 2 pi r.

131
00:08:37,950 --> 00:08:40,240
So 2 pi r is the circumference.

132
00:08:40,240 --> 00:08:42,710
And then the height is the
height of that green stem

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00:08:42,710 --> 00:08:43,210
there.

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00:08:43,210 --> 00:08:45,460
That's e^(-r^2).

135
00:08:45,460 --> 00:08:50,460
And then we're multiplying by
the thickness, which is dr.

136
00:08:50,460 --> 00:08:52,530
So the thickness
of the green is dr.

137
00:08:52,530 --> 00:08:57,860
The height of this little
green guy is e^(-r^2),

138
00:08:57,860 --> 00:09:01,310
and the circumference is
2 pi times the radius,

139
00:09:01,310 --> 00:09:03,701
when we sweep the circle around.

140
00:09:03,701 --> 00:09:04,200
Question.

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00:09:04,200 --> 00:09:21,970
STUDENT: [INAUDIBLE]

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00:09:21,970 --> 00:09:23,850
PROFESSOR: So the
question is, why

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00:09:23,850 --> 00:09:28,020
is what we're interested
in not this pink area.

144
00:09:28,020 --> 00:09:31,810
And the reason is
an interpretation

145
00:09:31,810 --> 00:09:33,430
of what I meant by this.

146
00:09:33,430 --> 00:09:38,310
What I meant by this is
that if you wanted to add up

147
00:09:38,310 --> 00:09:42,190
what the likelihood is that this
thing will be here versus here,

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00:09:42,190 --> 00:09:44,940
I want it to be,
really, the proportions

149
00:09:44,940 --> 00:09:47,780
are the number of hits
times the, if you like,

150
00:09:47,780 --> 00:09:49,590
I wanted it d area.

151
00:09:49,590 --> 00:09:51,850
That's really what I meant here.

152
00:09:51,850 --> 00:09:56,740
The number of hits
in a little chunk.

153
00:09:56,740 --> 00:10:01,370
So little, maybe I'll call
it delta A. A little chunk.

154
00:10:01,370 --> 00:10:04,480
Is proportional to
the chunk times that.

155
00:10:04,480 --> 00:10:05,920
So there's already
an area factor.

156
00:10:05,920 --> 00:10:06,920
And there's a height.

157
00:10:06,920 --> 00:10:09,580
So there are a total of
three dimensions involved.

158
00:10:09,580 --> 00:10:13,120
There's the area and
then this height.

159
00:10:13,120 --> 00:10:18,640
So it's a matter of the--
what I was given, what I

160
00:10:18,640 --> 00:10:21,720
intended to say the problem is.

161
00:10:21,720 --> 00:10:22,720
Yeah, another question.

162
00:10:22,720 --> 00:10:25,084
STUDENT: [INAUDIBLE]

163
00:10:25,084 --> 00:10:26,000
PROFESSOR: Yeah, yeah.

164
00:10:26,000 --> 00:10:27,060
Exactly.

165
00:10:27,060 --> 00:10:30,570
The height is c times
that, whatever this c is.

166
00:10:30,570 --> 00:10:32,220
In fact, we don't
know what the c is,

167
00:10:32,220 --> 00:10:34,460
but because we're going to
have a part and a whole,

168
00:10:34,460 --> 00:10:37,490
we'll divide, the c
will always cancel.

169
00:10:37,490 --> 00:10:39,240
So I'm throwing the c out.

170
00:10:39,240 --> 00:10:42,750
I don't know what it is, and
in the end it won't matter.

171
00:10:42,750 --> 00:10:44,330
That's a very
important question.

172
00:10:44,330 --> 00:10:44,830
Yes.

173
00:10:44,830 --> 00:10:49,180
STUDENT: [INAUDIBLE]

174
00:10:49,180 --> 00:10:50,180
PROFESSOR: Say it again?

175
00:10:50,180 --> 00:11:02,040
STUDENT: PROFESSOR:

176
00:11:02,040 --> 00:11:05,920
PROFESSOR: So what I mean is the
number of hits in some chunk.

177
00:11:05,920 --> 00:11:09,520
That is, suppose you
imagine, the question

178
00:11:09,520 --> 00:11:11,720
is, what does this
left-hand side mean.

179
00:11:11,720 --> 00:11:12,250
That right?

180
00:11:12,250 --> 00:11:17,690
Is that the question
that's you're asking?

181
00:11:17,690 --> 00:11:21,040
When I try to understand
what the distribution

182
00:11:21,040 --> 00:11:25,420
of dartboard hits is, I
should imagine my dartboard.

183
00:11:25,420 --> 00:11:28,400
And very often there'll
be a whole bunch

184
00:11:28,400 --> 00:11:29,700
of holes in some places.

185
00:11:29,700 --> 00:11:30,810
And fewer holes else.

186
00:11:30,810 --> 00:11:33,010
I'm trying to figure out
what the whole distribution

187
00:11:33,010 --> 00:11:35,590
of those marks is.

188
00:11:35,590 --> 00:11:38,410
And so some places will
have more hits on them

189
00:11:38,410 --> 00:11:41,060
and some places will
have fewer hits on them.

190
00:11:41,060 --> 00:11:43,980
And so what I want to
measure is, on average,

191
00:11:43,980 --> 00:11:44,910
the number of hits.

192
00:11:44,910 --> 00:11:47,387
So this would really
be some-- This constant

193
00:11:47,387 --> 00:11:49,470
of proportionality is
ambiguous because it depends

194
00:11:49,470 --> 00:11:50,553
on how many times you try.

195
00:11:50,553 --> 00:11:52,470
If you throw a
thousand times, it'll

196
00:11:52,470 --> 00:11:54,320
be much more densely packed.

197
00:11:54,320 --> 00:11:58,780
And if you have only a
hundred times it'll be fewer.

198
00:11:58,780 --> 00:12:01,850
So that's where this
constant comes in.

199
00:12:01,850 --> 00:12:04,610
But given that you have
a certain number of times

200
00:12:04,610 --> 00:12:06,790
that you tried, say,
a thousand times,

201
00:12:06,790 --> 00:12:09,500
there will be a whole bunch
more piled in the middle.

202
00:12:09,500 --> 00:12:12,020
And fewer and fewer as
you get farther away.

203
00:12:12,020 --> 00:12:15,800
Assuming that the person's
aim is reasonable.

204
00:12:15,800 --> 00:12:17,024
So that's what we're saying.

205
00:12:17,024 --> 00:12:19,440
So we're thinking in terms
of-- The person's always aiming

206
00:12:19,440 --> 00:12:20,065
for the center.

207
00:12:20,065 --> 00:12:22,660
So it's most likely that the
person will hit the center.

208
00:12:22,660 --> 00:12:24,820
But on the other hand,
it's a fallible person,

209
00:12:24,820 --> 00:12:26,440
so the person may miss.

210
00:12:26,440 --> 00:12:29,272
And so it's less and less
likely as you go farther out.

211
00:12:29,272 --> 00:12:31,480
And we're just counting how
many times this gets hit,

212
00:12:31,480 --> 00:12:33,220
how many time this and so on.

213
00:12:33,220 --> 00:12:34,830
In proportion to the area.

214
00:12:34,830 --> 00:12:40,460
STUDENT: [INAUDIBLE]

215
00:12:40,460 --> 00:12:42,252
PROFESSOR: Yeah. r_1
and r_2 are arbitrary.

216
00:12:42,252 --> 00:12:44,251
We're going to make this
calculation in general.

217
00:12:44,251 --> 00:12:46,240
We're going to calculate
what the likelihood is

218
00:12:46,240 --> 00:12:50,290
that we hit any possible band.

219
00:12:50,290 --> 00:12:53,010
And I want to leave those
as just letters for now.

220
00:12:53,010 --> 00:12:54,990
The r_1 and the r_2.

221
00:12:54,990 --> 00:12:57,110
Because I want to be able
to try various different

222
00:12:57,110 --> 00:12:57,990
possibilities.

223
00:12:57,990 --> 00:13:03,730
STUDENT: [INAUDIBLE]

224
00:13:03,730 --> 00:13:06,310
PROFESSOR: Say it again, why
do we have to take volume?

225
00:13:06,310 --> 00:13:08,830
So this is what we
were addressing before.

226
00:13:08,830 --> 00:13:13,900
It's a volume because it's
number of hit per unit area.

227
00:13:13,900 --> 00:13:17,590
So there's a height,
that is, number of hits.

228
00:13:17,590 --> 00:13:19,820
And then there's an area
and the product of those

229
00:13:19,820 --> 00:13:22,610
is-- So this is, if
you like, a histogram

230
00:13:22,610 --> 00:13:25,100
of the number of hits.

231
00:13:25,100 --> 00:13:26,910
But this should be
measured per area.

232
00:13:26,910 --> 00:13:29,400
Not per length of r.

233
00:13:29,400 --> 00:13:32,020
Because on the real diagram,
it's going all the way around.

234
00:13:32,020 --> 00:13:35,760
There's a lot more
area to this red band

235
00:13:35,760 --> 00:13:41,150
than just the distance implies.

236
00:13:41,150 --> 00:13:42,940
OK.

237
00:13:42,940 --> 00:13:47,460
So, having discussed
the setup, this

238
00:13:47,460 --> 00:13:49,770
is a pretty standard setup--
Oh, one more question.

239
00:13:49,770 --> 00:13:50,270
Yes.

240
00:13:50,270 --> 00:14:13,564
STUDENT: [INAUDIBLE]

241
00:14:13,564 --> 00:14:14,230
PROFESSOR: Yeah.

242
00:14:14,230 --> 00:14:17,700
So the question is, why
is this a realistic.

243
00:14:17,700 --> 00:14:21,260
Why is this choice of
function here, e^(-r^2),

244
00:14:21,260 --> 00:14:25,670
a realistic choice of
function for the darts.

245
00:14:25,670 --> 00:14:30,480
So I can answer this
with an analogy.

246
00:14:30,480 --> 00:14:32,800
When people were
asking themselves

247
00:14:32,800 --> 00:14:38,120
where the V-2 rockets
from Germany hit London,

248
00:14:38,120 --> 00:14:39,170
they used this model.

249
00:14:39,170 --> 00:14:43,100
It turned out to be the one
which was the most accurate.

250
00:14:43,100 --> 00:14:48,630
So that gives you an idea
that this is actually real.

251
00:14:48,630 --> 00:14:53,670
The question, this makes it
look like people are masters.

252
00:14:53,670 --> 00:14:56,940
That is, that they'll
all hit in the center

253
00:14:56,940 --> 00:14:58,870
more often than elsewhere.

254
00:14:58,870 --> 00:15:03,960
But that's actually
somewhat deceptive.

255
00:15:03,960 --> 00:15:08,040
There's a difference
between the mode,

256
00:15:08,040 --> 00:15:13,510
that is, the most likely spot,
and what happens on average.

257
00:15:13,510 --> 00:15:17,960
So in other words, the single
most likely spot is the center.

258
00:15:17,960 --> 00:15:20,720
But there's rather little
area in here, and in fact

259
00:15:20,720 --> 00:15:24,860
the likelihood of hitting
that is some little tiny bit.

260
00:15:24,860 --> 00:15:25,620
In here.

261
00:15:25,620 --> 00:15:29,850
In fact, you're much more
likely to be out here.

262
00:15:29,850 --> 00:15:31,539
So if you take the
total of the volume,

263
00:15:31,539 --> 00:15:32,955
you'll see that
much of the volume

264
00:15:32,955 --> 00:15:34,260
is contributed from out here.

265
00:15:34,260 --> 00:15:37,780
And in fact, the person hits
rather rarely near the center.

266
00:15:37,780 --> 00:15:42,737
So this is not a
ridiculous thing to do.

267
00:15:42,737 --> 00:15:45,320
If you think of it in terms of
somebody's aiming at the center

268
00:15:45,320 --> 00:15:46,950
but there's some
random thing which

269
00:15:46,950 --> 00:15:49,770
is throwing the
person off, then there

270
00:15:49,770 --> 00:15:51,710
is likely to be to
left or to the right,

271
00:15:51,710 --> 00:15:54,080
or they might even get
lucky and all those errors

272
00:15:54,080 --> 00:15:55,470
cancel themselves
and they happen

273
00:15:55,470 --> 00:15:59,090
to hit pretty much
near the center.

274
00:15:59,090 --> 00:16:00,240
Yeah, another question.

275
00:16:00,240 --> 00:16:06,266
STUDENT: [INAUDIBLE] PROFESSOR:
How does the little brother

276
00:16:06,266 --> 00:16:06,890
come into play?

277
00:16:06,890 --> 00:16:09,390
The little brother is going
to come into play as follows.

278
00:16:09,390 --> 00:16:11,300
I'll tell you in advance.

279
00:16:11,300 --> 00:16:15,500
So the thing is, the little
brother was not so stupid

280
00:16:15,500 --> 00:16:19,760
as to stand in
front of the target.

281
00:16:19,760 --> 00:16:22,220
I know.

282
00:16:22,220 --> 00:16:26,800
He stood about twice the
radius of the target away.

283
00:16:26,800 --> 00:16:29,960
And so, we're going to
approximate the location

284
00:16:29,960 --> 00:16:32,460
by some sector here.

285
00:16:32,460 --> 00:16:35,920
Which is just going to be some
chunk of one of these things.

286
00:16:35,920 --> 00:16:38,010
We'll just break
off a piece of it.

287
00:16:38,010 --> 00:16:39,850
And that's how we're
going to capture.

288
00:16:39,850 --> 00:16:42,030
So the point is,
the target is here.

289
00:16:42,030 --> 00:16:46,000
But there is the possibility
that the seven-year-old

290
00:16:46,000 --> 00:16:48,550
who's throwing the darts
actually missed the target.

291
00:16:48,550 --> 00:16:50,780
That actually happens a lot.

292
00:16:50,780 --> 00:16:54,660
So, does that answer
your question?

293
00:16:54,660 --> 00:16:55,530
Alright.

294
00:16:55,530 --> 00:16:57,170
Are we ready now to to do this?

295
00:16:57,170 --> 00:16:57,950
One more question.

296
00:16:57,950 --> 00:17:05,250
STUDENT: [INAUDIBLE]

297
00:17:05,250 --> 00:17:08,320
PROFESSOR: I'm giving
you this property here.

298
00:17:08,320 --> 00:17:10,110
I'm telling you
that-- This is what's

299
00:17:10,110 --> 00:17:12,540
called a mathematical
model, when you give

300
00:17:12,540 --> 00:17:13,900
somebody something like this.

301
00:17:13,900 --> 00:17:16,450
In fact, that requires
further justification.

302
00:17:16,450 --> 00:17:19,120
It's an interesting issue.

303
00:17:19,120 --> 00:17:19,620
Yeah.

304
00:17:19,620 --> 00:17:26,084
STUDENT: [INAUDIBLE]

305
00:17:26,084 --> 00:17:27,750
PROFESSOR: I'm giving
it to you for now.

306
00:17:27,750 --> 00:17:30,120
And it's something which
really has to be justified.

307
00:17:30,120 --> 00:17:32,080
In certain circumstances
it is justified.

308
00:17:32,080 --> 00:17:37,620
But, OK.

309
00:17:37,620 --> 00:17:43,800
So anyway, here's our part.

310
00:17:43,800 --> 00:17:46,200
This is going to be
our chunk, for now,

311
00:17:46,200 --> 00:17:50,760
that we're going to
estimate the importance,

312
00:17:50,760 --> 00:17:56,210
the relative importance, of.

313
00:17:56,210 --> 00:18:01,460
And now, this is something whose
antiderivative we can just do

314
00:18:01,460 --> 00:18:03,600
by substitution or by guessing.

315
00:18:03,600 --> 00:18:07,870
It's just -pi e^(-r^2).

316
00:18:07,870 --> 00:18:10,650
If you differentiate
that, you get a -2r, which

317
00:18:10,650 --> 00:18:12,730
cancels the minus sign here.

318
00:18:12,730 --> 00:18:15,370
So you get 2pi r e^(-r^2).

319
00:18:15,370 --> 00:18:18,400
So that's the antiderivative.

320
00:18:18,400 --> 00:18:24,710
And we're evaluating
it at r_1 and r_2.

321
00:18:24,710 --> 00:18:28,490
So with the minus sign
that's going to get reversed.

322
00:18:28,490 --> 00:18:33,350
The answer is going to
be pi times e^(-r_1^2),

323
00:18:33,350 --> 00:18:39,090
that's the bottom one, minus
e^(r_2^2), that's the top.

324
00:18:39,090 --> 00:18:45,777
So this is what
our part gives us.

325
00:18:45,777 --> 00:18:48,110
And, more technically, if you
wanted to multiply through

326
00:18:48,110 --> 00:18:50,350
by c, it would be c times this.

327
00:18:50,350 --> 00:18:55,140
I'll say that in just a second.

328
00:18:55,140 --> 00:18:59,280
OK, now I want to
work-- So, if you like,

329
00:18:59,280 --> 00:19:06,490
the part is equal to, maybe
even I'll call it c pi,

330
00:19:06,490 --> 00:19:07,740
times e^(-r_1^2) - e^(-r_2^2).

331
00:19:12,429 --> 00:19:14,470
That's what it really is
if I put in this factor.

332
00:19:14,470 --> 00:19:17,690
So now there's no prejudice as
to how many attempts we make.

333
00:19:17,690 --> 00:19:20,490
Whether it was a thousand
attempts or a million

334
00:19:20,490 --> 00:19:22,740
attempts at the target.

335
00:19:22,740 --> 00:19:25,480
Now, the most important
second feature

336
00:19:25,480 --> 00:19:27,250
here of these kinds
of modeling problems

337
00:19:27,250 --> 00:19:30,560
is, there is always some
kind of idealization.

338
00:19:30,560 --> 00:19:33,070
And the next thing that I
want to discuss with you

339
00:19:33,070 --> 00:19:36,940
is the interpretation
of the whole.

340
00:19:36,940 --> 00:19:41,870
That is, what's the family
of all possibilities.

341
00:19:41,870 --> 00:19:45,790
And in this case, what
I'm going to claim

342
00:19:45,790 --> 00:19:50,160
is that the reasonable
way to think of the whole

343
00:19:50,160 --> 00:19:55,940
is it's that r can range all
the way from 0 to infinity.

344
00:19:55,940 --> 00:19:57,210
Now, you may not like this.

345
00:19:57,210 --> 00:20:01,730
But these are maybe my first
and third favorite number,

346
00:20:01,730 --> 00:20:03,810
my second favorite
number being 1.

347
00:20:03,810 --> 00:20:07,930
So infinity is a
really useful concept.

348
00:20:07,930 --> 00:20:12,010
Of course, it's nonsense in
the context of the darts.

349
00:20:12,010 --> 00:20:14,770
Because if you think
of the basement wall

350
00:20:14,770 --> 00:20:17,140
where the kid might
miss a target,

351
00:20:17,140 --> 00:20:19,619
he'd probably hit the wall.

352
00:20:19,619 --> 00:20:22,160
He's probably not going to hit
one of the walls to the right,

353
00:20:22,160 --> 00:20:25,080
and anyway he's certainly
not going to hit over there.

354
00:20:25,080 --> 00:20:28,870
So there's something
artificial about sending

355
00:20:28,870 --> 00:20:32,390
the possibility of hits all
the way out to infinity.

356
00:20:32,390 --> 00:20:34,850
On the other hand, the
shape of this curve

357
00:20:34,850 --> 00:20:38,310
is such that the
real tail ends here,

358
00:20:38,310 --> 00:20:40,990
because of this exponential
decrease, are tiny.

359
00:20:40,990 --> 00:20:42,630
And that's negligible.

360
00:20:42,630 --> 00:20:45,700
And the point is that
actually the value,

361
00:20:45,700 --> 00:20:47,550
if you go all the
way out to infinity,

362
00:20:47,550 --> 00:20:50,020
is the easiest
value to calculate.

363
00:20:50,020 --> 00:20:52,697
So by doing this, I'm
idealizing the problem

364
00:20:52,697 --> 00:20:54,530
but I'm actually making
the numbers come out

365
00:20:54,530 --> 00:20:55,890
much more cleanly.

366
00:20:55,890 --> 00:20:58,180
And this is just always
done in mathematics.

367
00:20:58,180 --> 00:21:00,290
That's what we did when
we went from differences

368
00:21:00,290 --> 00:21:02,460
to differentials,
to differentiation

369
00:21:02,460 --> 00:21:04,640
and infinitesimals.

370
00:21:04,640 --> 00:21:06,282
So we like that.

371
00:21:06,282 --> 00:21:08,740
Because it makes things easier,
not because it makes things

372
00:21:08,740 --> 00:21:10,320
harder.

373
00:21:10,320 --> 00:21:12,240
So we're just going to
pretend the whole is

374
00:21:12,240 --> 00:21:13,450
from 0 to infinity.

375
00:21:13,450 --> 00:21:15,550
And now let's just
see what it is.

376
00:21:15,550 --> 00:21:21,330
It's c pi times, the
starting place is e^(-0^2).

377
00:21:21,330 --> 00:21:24,630
That's r_1, right, this is the
role that r_1 plays is this,

378
00:21:24,630 --> 00:21:26,374
and the r_2 is this value.

379
00:21:26,374 --> 00:21:27,290
Minus e^(-infinity^2).

380
00:21:30,180 --> 00:21:33,940
Which is that negligibly
small number, 0.

381
00:21:33,940 --> 00:21:36,750
So this is just c pi.

382
00:21:36,750 --> 00:21:39,700
Because this number is 1,
and this other number is 0.

383
00:21:39,700 --> 00:21:48,420
This is just (1 - 0), in
the parentheses there.

384
00:21:48,420 --> 00:21:52,300
And now I can tell you
from these two numbers

385
00:21:52,300 --> 00:21:54,990
what the probability is.

386
00:21:54,990 --> 00:21:58,450
The probability that
we landed on the target

387
00:21:58,450 --> 00:22:04,250
in a radius between r_1 and r_2,
so that's this annulus here,

388
00:22:04,250 --> 00:22:12,350
is the ratio of the
part to the whole.

389
00:22:12,350 --> 00:22:16,060
Which in this case
just cancels the c pi.

390
00:22:16,060 --> 00:22:21,280
So it's e^(-r_1^2) - e^(-r_2^2).

391
00:22:21,280 --> 00:22:25,520
There's the formula
for the probability.

392
00:22:25,520 --> 00:22:28,100
So the c canceled
and the pi canceled.

393
00:22:28,100 --> 00:22:32,170
It's all gone.

394
00:22:32,170 --> 00:22:34,040
Now, again, let
me just emphasize

395
00:22:34,040 --> 00:22:35,750
the way this formula
is supposed to work.

396
00:22:35,750 --> 00:22:42,210
The total probability
of every possibility

397
00:22:42,210 --> 00:22:45,900
here is supposed to
be set up to be 1.

398
00:22:45,900 --> 00:22:48,530
This is some fraction of 1.

399
00:22:48,530 --> 00:22:51,191
If you like, it's a percent.

400
00:22:51,191 --> 00:22:51,690
Yes.

401
00:22:51,690 --> 00:22:57,149
STUDENT: [INAUDIBLE]

402
00:22:57,149 --> 00:22:59,690
PROFESSOR: This one is giving--
The question is, doesn't this

403
00:22:59,690 --> 00:23:00,780
just give the
probability of the ring?

404
00:23:00,780 --> 00:23:02,620
This gives you the
probability of the ring,

405
00:23:02,620 --> 00:23:05,510
but this is a very,
very wide ring.

406
00:23:05,510 --> 00:23:08,479
This is a ring starting with
0, nothing, on the inside.

407
00:23:08,479 --> 00:23:09,770
And then going all the way out.

408
00:23:09,770 --> 00:23:11,880
So that's everything.

409
00:23:11,880 --> 00:23:13,380
So this corresponds
to everything.

410
00:23:13,380 --> 00:23:22,650
This corresponds to a ring.

411
00:23:22,650 --> 00:23:24,990
So now, let's see.

412
00:23:24,990 --> 00:23:26,390
Where do I want to go from here.

413
00:23:26,390 --> 00:23:28,550
So in order to
make progress here,

414
00:23:28,550 --> 00:23:31,230
I still have to give you one
more piece of information.

415
00:23:31,230 --> 00:23:35,650
And this is, again,
supposed to be realistic.

416
00:23:35,650 --> 00:23:40,350
When I was three years old
and my brother's friend Ralph

417
00:23:40,350 --> 00:23:43,920
was seven, I watched him
throwing darts a lot.

418
00:23:43,920 --> 00:23:54,200
And I would say that for Ralph,
so for Ralph, at age seven,

419
00:23:54,200 --> 00:23:57,080
anyway, later on he got
a little better at it.

420
00:23:57,080 --> 00:23:59,950
But Ralph at age seven,
the probability that he

421
00:23:59,950 --> 00:24:08,440
hit the target was about 1/2.

422
00:24:08,440 --> 00:24:09,660
Right?

423
00:24:09,660 --> 00:24:12,040
So he hit the target
about half the time.

424
00:24:12,040 --> 00:24:14,560
And the other times,
there was cement

425
00:24:14,560 --> 00:24:17,870
on the walls of the
basement, it wasn't that bad.

426
00:24:17,870 --> 00:24:19,250
Just bounced off.

427
00:24:19,250 --> 00:24:22,000
That also meant that the points
got a little blunter as time

428
00:24:22,000 --> 00:24:22,560
went on.

429
00:24:22,560 --> 00:24:26,150
So it was a little less
dangerous when they hit.

430
00:24:26,150 --> 00:24:26,650
Alright.

431
00:24:26,650 --> 00:24:33,210
So now, so here's the extra
assumption that I want to make.

432
00:24:33,210 --> 00:24:41,260
So a is going to be the
radius of the target.

433
00:24:41,260 --> 00:24:45,660
Now, the other realistic
assumption that I want to make

434
00:24:45,660 --> 00:24:49,170
is where this little
kid would be standing.

435
00:24:49,170 --> 00:24:51,650
And now, here, I want
to get very specific

436
00:24:51,650 --> 00:24:54,280
and just do the
computation in one case.

437
00:24:54,280 --> 00:25:00,430
We're going to imagine
the target is here.

438
00:25:00,430 --> 00:25:03,200
And the kid is standing,
say, between-- So we'll just

439
00:25:03,200 --> 00:25:04,430
do a section of this.

440
00:25:04,430 --> 00:25:13,130
This is between 3
o'clock and 5 o'clock.

441
00:25:13,130 --> 00:25:16,740
There's more of him, but it's
lower down and maybe negligible

442
00:25:16,740 --> 00:25:17,350
here.

443
00:25:17,350 --> 00:25:20,400
So this section is
the part, the chunk

444
00:25:20,400 --> 00:25:23,330
that we want to see about.

445
00:25:23,330 --> 00:25:31,210
And this is a, and then
this distance here is 2a.

446
00:25:31,210 --> 00:25:35,330
And then the longest
distance here is 3a.

447
00:25:35,330 --> 00:25:39,000
So the longest distance is 3a.

448
00:25:39,000 --> 00:25:40,870
So in other words,
what I'm saying

449
00:25:40,870 --> 00:25:50,180
is that the probability, if
you're standing too close,

450
00:25:50,180 --> 00:26:06,660
the chance Ralph hits younger
brother is about 1/6-- Right?

451
00:26:06,660 --> 00:26:12,820
Because 2/12 = 1/6.

452
00:26:12,820 --> 00:26:22,020
1/6 of the probability that
we're between a and 3a.

453
00:26:22,020 --> 00:26:29,052
That's the number that
we're looking for.

454
00:26:29,052 --> 00:26:29,760
Another question.

455
00:26:29,760 --> 00:26:33,900
STUDENT: [INAUDIBLE]

456
00:26:33,900 --> 00:26:36,680
PROFESSOR: The 2/12 came from
the fact that we assumed.

457
00:26:36,680 --> 00:26:39,250
So we made a very, very
bold assumption here.

458
00:26:39,250 --> 00:26:41,530
We assumed that
this human being,

459
00:26:41,530 --> 00:26:46,760
who is actually standing--
The floor is about down here.

460
00:26:46,760 --> 00:26:48,460
Maybe he wasn't that tall.

461
00:26:48,460 --> 00:26:51,560
But anyway, so he's really
a little bigger than this.

462
00:26:51,560 --> 00:26:53,850
That the part of him that
was close to the target

463
00:26:53,850 --> 00:27:00,500
covered about this section
here, between radius 2a and 3a.

464
00:27:00,500 --> 00:27:02,740
As you'll see, actually
from the computation,

465
00:27:02,740 --> 00:27:07,290
because the likelihood drops
off pretty quickly, whatever

466
00:27:07,290 --> 00:27:09,140
of him was standing
outside there

467
00:27:09,140 --> 00:27:11,269
wouldn't have mattered anyway.

468
00:27:11,269 --> 00:27:13,060
So we're just worried
about the part that's

469
00:27:13,060 --> 00:27:18,677
closest to the target here.

470
00:27:18,677 --> 00:27:19,510
STUDENT: [INAUDIBLE]

471
00:27:19,510 --> 00:27:20,801
PROFESSOR: Why is it out of 12?

472
00:27:20,801 --> 00:27:22,650
Because I made it a clock.

473
00:27:22,650 --> 00:27:25,110
And I made it from 3
o'clock to 5 o'clock,

474
00:27:25,110 --> 00:27:28,790
so it's 2 of the 12
hours of a clock.

475
00:27:28,790 --> 00:27:32,640
It's just a way of me making
a section that you can visibly

476
00:27:32,640 --> 00:27:36,400
see.

477
00:27:36,400 --> 00:27:40,060
So now, so here's what
we're trying to calculate.

478
00:27:40,060 --> 00:27:47,790
And in order to figure this
out, I need one more item. here.

479
00:27:47,790 --> 00:27:49,870
So maybe I'll leave myself
a little bit of room.

480
00:27:49,870 --> 00:27:52,010
I have to figure out
something about what

481
00:27:52,010 --> 00:27:53,830
our information gave us.

482
00:27:53,830 --> 00:28:00,380
Which is that the probability,
sorry, this probability

483
00:28:00,380 --> 00:28:01,690
was 1/2.

484
00:28:01,690 --> 00:28:03,580
So let's remember what this is.

485
00:28:03,580 --> 00:28:09,470
This is going to be
e^(-0^2) - e^(-a^2).

486
00:28:09,470 --> 00:28:11,180
That's what's this
probability is.

487
00:28:11,180 --> 00:28:16,210
And that's equal to 1/2.

488
00:28:16,210 --> 00:28:21,330
So that means that -
1 - e^(-a^2). = 1/2.

489
00:28:21,330 --> 00:28:26,620
Which means that e^(-a^2) = 1/2.

490
00:28:26,620 --> 00:28:28,220
I'm not going to
calculate a, this

491
00:28:28,220 --> 00:28:30,050
is the part about the
information about a

492
00:28:30,050 --> 00:28:32,480
that I want to use.

493
00:28:32,480 --> 00:28:33,520
That's what I'll use.

494
00:28:33,520 --> 00:28:36,250
And now I'm going to calculate
this other probability here.

495
00:28:36,250 --> 00:28:41,590
So the probability
right up there is this.

496
00:28:41,590 --> 00:28:47,390
And that's going to be the same
as e^(-(2a)^2) - e^(-(3a)^2).

497
00:28:51,610 --> 00:28:54,650
That's what we calculated.

498
00:28:54,650 --> 00:28:59,200
And now I want to use some of
the arithmetic of exponents.

499
00:28:59,200 --> 00:29:01,380
This is (e^(-a^2))^2.

500
00:29:05,180 --> 00:29:11,890
Because it's really (2a)^2
is 4-- the quantity is 4a^2,

501
00:29:11,890 --> 00:29:14,580
and then I bring
that exponent out.

502
00:29:14,580 --> 00:29:19,740
Minus e^(-a^2) to the
9th power, that's 3^2.

503
00:29:25,040 --> 00:29:28,560
And so this comes out
to be (1/2)^4 - (1/2)^9.

504
00:29:36,190 --> 00:29:39,340
Which is approximately 1/16.

505
00:29:39,340 --> 00:29:40,380
This is negligible.

506
00:29:40,380 --> 00:29:41,180
This part here.

507
00:29:41,180 --> 00:29:43,110
And this is actually
why these tails,

508
00:29:43,110 --> 00:29:48,130
as you go out to infinity,
don't really matter that much.

509
00:29:48,130 --> 00:29:49,880
So this is a much
smaller number.

510
00:29:49,880 --> 00:29:53,390
So the probability of
the whole band is 1/16.

511
00:29:53,390 --> 00:29:55,720
And now I can answer
the question up here.

512
00:29:55,720 --> 00:30:01,110
This is approximately
1/6 * 1/16.

513
00:30:01,110 --> 00:30:07,340
Which is about
1/100, or about 1%.

514
00:30:07,340 --> 00:30:13,340
So if I stood there
for 100 attempts,

515
00:30:13,340 --> 00:30:18,720
then my chances of getting
hit were pretty high.

516
00:30:18,720 --> 00:30:27,970
So that's the computation.

517
00:30:27,970 --> 00:30:34,940
That's a typical example of
a problem in probability.

518
00:30:34,940 --> 00:30:36,860
And let me just make
one more connection

519
00:30:36,860 --> 00:30:42,660
with what we did before.

520
00:30:42,660 --> 00:30:46,130
This is connected to weighted
averages or integrals

521
00:30:46,130 --> 00:30:47,470
over weights.

522
00:30:47,470 --> 00:30:53,110
But the weight that's
involved in this problem

523
00:30:53,110 --> 00:30:59,810
was w(r) is equal to-- So let's
just look at what happened

524
00:30:59,810 --> 00:31:01,520
in all of those integrals.

525
00:31:01,520 --> 00:31:03,790
What happened in all
the integrals was, we

526
00:31:03,790 --> 00:31:09,060
had this factor here, 2 pi r.

527
00:31:09,060 --> 00:31:23,330
And if I include this c, it
was really 2 pi c r e^(-r^2).

528
00:31:23,330 --> 00:31:27,960
This was the weight
that we were using.

529
00:31:27,960 --> 00:31:30,770
The relative
importance of things.

530
00:31:30,770 --> 00:31:34,260
Now, this is not the same as
the e^(-r^2) that we started out

531
00:31:34,260 --> 00:31:34,760
with.

532
00:31:34,760 --> 00:31:38,000
Because this is the
one-dimensional histogram.

533
00:31:38,000 --> 00:31:40,850
And that has to do with the
method of shells that gave us

534
00:31:40,850 --> 00:31:43,210
that extra factor of r here.

535
00:31:43,210 --> 00:31:45,090
So that also connects
with the question

536
00:31:45,090 --> 00:31:47,610
at the beginning, which had
to do with this paradox,

537
00:31:47,610 --> 00:31:50,420
that it looks like these
places in the middle

538
00:31:50,420 --> 00:31:51,440
are the most likely.

539
00:31:51,440 --> 00:31:54,500
But that's the plot e^(-r^2).

540
00:31:54,500 --> 00:31:57,240
If you actually look
at this plot here,

541
00:31:57,240 --> 00:32:00,330
you see that as r goes
to 0, it's going to 0.

542
00:32:00,330 --> 00:32:03,010
This is a different graph here.

543
00:32:03,010 --> 00:32:07,720
And actually, so this is
what's happening really.

544
00:32:07,720 --> 00:32:09,730
In terms of how likely
it is that you'll

545
00:32:09,730 --> 00:32:18,392
get within a certain distance
of the center of the target.

546
00:32:18,392 --> 00:32:20,850
Again, it's not the area under
that curve that we're doing.

547
00:32:20,850 --> 00:32:27,940
It's that volume of revolution.

548
00:32:27,940 --> 00:32:29,494
We're going to
change subjects now.

549
00:32:29,494 --> 00:32:30,410
OK, one more question.

550
00:32:30,410 --> 00:32:31,937
Yes.

551
00:32:31,937 --> 00:32:32,770
STUDENT: [INAUDIBLE]

552
00:32:32,770 --> 00:32:36,120
PROFESSOR: Yeah, that's supposed
to be the graph of w(r).

553
00:32:36,120 --> 00:32:47,359
STUDENT: [INAUDIBLE]

554
00:32:47,359 --> 00:32:48,900
PROFESSOR: Well,
so, the question is,

555
00:32:48,900 --> 00:32:50,983
wouldn't the importance
of the center be greatest?

556
00:32:50,983 --> 00:32:53,790
It's a question of which
variable you're using.

557
00:32:53,790 --> 00:32:57,057
According to pure
radius, it's not.

558
00:32:57,057 --> 00:32:58,640
It turns out that
there are some bands

559
00:32:58,640 --> 00:33:00,930
in radius which are more
important, more likely

560
00:33:00,930 --> 00:33:06,206
for hits than others.

561
00:33:06,206 --> 00:33:07,580
It really has to
do with the fact

562
00:33:07,580 --> 00:33:10,980
that the center, or the core,
of the target is really tiny.

563
00:33:10,980 --> 00:33:11,950
So it's harder to hit.

564
00:33:11,950 --> 00:33:16,170
Whereas a whole band around the
outside has a lot more area.

565
00:33:16,170 --> 00:33:18,500
Many, many ways
to hit that band.

566
00:33:18,500 --> 00:33:19,810
So it's a much larger area.

567
00:33:19,810 --> 00:33:24,220
So there's a competition there
between those two things.

568
00:33:24,220 --> 00:33:25,910
So we're going to move on.

569
00:33:25,910 --> 00:33:28,170
And I want to talk now
about a different kind

570
00:33:28,170 --> 00:33:30,550
of weighted average.

571
00:33:30,550 --> 00:33:36,410
These weighted averages are
going to be much simpler.

572
00:33:36,410 --> 00:34:01,260
And they come up in what's
called numerical integration.

573
00:34:01,260 --> 00:34:04,597
There are many, many methods
of integrating numerically.

574
00:34:04,597 --> 00:34:06,680
And they're important
because many, many integrals

575
00:34:06,680 --> 00:34:08,510
don't have formulas.

576
00:34:08,510 --> 00:34:13,980
And so you have to compute them
with a calculator or a machine.

577
00:34:13,980 --> 00:34:24,400
So the first type that we've
already done are Riemann sums.

578
00:34:24,400 --> 00:34:28,170
They turned out to be
incredibly inefficient.

579
00:34:28,170 --> 00:34:30,270
They're lousy.

580
00:34:30,270 --> 00:34:32,120
The next rule that
I'm going to describe

581
00:34:32,120 --> 00:34:33,370
is a little improvement.

582
00:34:33,370 --> 00:34:38,130
It's called the
trapezoidal rule.

583
00:34:38,130 --> 00:34:41,340
And this one is
much more reasonable

584
00:34:41,340 --> 00:34:43,370
than the Riemann sum.

585
00:34:43,370 --> 00:34:49,140
Unfortunately, it's
actually also pretty lousy.

586
00:34:49,140 --> 00:34:53,150
There's another rule which is
just a slightly trickier rule.

587
00:34:53,150 --> 00:34:56,030
And it's actually
amazing that it exists.

588
00:34:56,030 --> 00:35:01,750
And it's called Simpson's Rule.

589
00:35:01,750 --> 00:35:06,690
And this one is
actually pretty good.

590
00:35:06,690 --> 00:35:07,770
It's clever.

591
00:35:07,770 --> 00:35:12,320
So let's get started with these.

592
00:35:12,320 --> 00:35:15,030
And the way I'll get
started is by reminding you

593
00:35:15,030 --> 00:35:16,030
what the Riemann sum is.

594
00:35:16,030 --> 00:35:17,779
So this is a good
review, because you need

595
00:35:17,779 --> 00:35:19,790
to know all three of these.

596
00:35:19,790 --> 00:35:26,540
And you're going to want to see
them all laid out in parallel.

597
00:35:26,540 --> 00:35:31,800
So here's the setup.

598
00:35:31,800 --> 00:35:33,580
OK, so here we go.

599
00:35:33,580 --> 00:35:36,660
We have our graph, we have our
function, it starts out at a,

600
00:35:36,660 --> 00:35:38,240
it ends up at b.

601
00:35:38,240 --> 00:35:40,160
Maybe goes on, but we're
only paying attention

602
00:35:40,160 --> 00:35:41,380
to this interval.

603
00:35:41,380 --> 00:35:44,760
This is a function y = f(x).

604
00:35:44,760 --> 00:35:48,160
And we split it up when
we do Riemann sums.

605
00:35:48,160 --> 00:35:50,450
So this is 1, this
is Riemann sums.

606
00:35:50,450 --> 00:35:53,930
We start out with a, which
is a point we call x_0,

607
00:35:53,930 --> 00:35:57,370
and then we go a
certain distance.

608
00:35:57,370 --> 00:35:58,710
We go all the way over to b.

609
00:35:58,710 --> 00:36:04,150
And we subdivide this
thing with these delta x's.

610
00:36:04,150 --> 00:36:08,130
Which are the step sizes.

611
00:36:08,130 --> 00:36:12,840
So we have these little
steps of size delta x.

612
00:36:12,840 --> 00:36:18,860
Corresponding to these x values,
we have y values. y_0 = f(x_0),

613
00:36:18,860 --> 00:36:21,090
that's the point above a.

614
00:36:21,090 --> 00:36:26,840
Then y_1 = f(x_1), that's
the point above x_1.

615
00:36:26,840 --> 00:36:33,120
And so forth, all the way
up to y_n, which is f(x_n).

616
00:36:33,120 --> 00:36:35,650
In order to figure out
the area, you really

617
00:36:35,650 --> 00:36:37,400
need to know something
about the function.

618
00:36:37,400 --> 00:36:39,217
You need to be able
to evaluate it.

619
00:36:39,217 --> 00:36:40,300
So that's what we've done.

620
00:36:40,300 --> 00:36:43,330
We've evaluated here
at n + 1 points.

621
00:36:43,330 --> 00:36:46,150
Enumerated 0 through n.

622
00:36:46,150 --> 00:36:48,910
And those are the
numbers out of which

623
00:36:48,910 --> 00:36:53,390
we're going to get all of our
approximations to the integral.

624
00:36:53,390 --> 00:36:56,370
So somehow we want
average these numbers.

625
00:36:56,370 --> 00:37:01,780
So here's our goal.

626
00:37:01,780 --> 00:37:14,450
Our goal is to average, or add,
I'm using average very loosely

627
00:37:14,450 --> 00:37:14,950
here.

628
00:37:14,950 --> 00:37:18,830
But I was going to say
add up these numbers

629
00:37:18,830 --> 00:37:25,750
to get an approximation.

630
00:37:25,750 --> 00:37:30,730
To average or add the y's.

631
00:37:30,730 --> 00:37:37,670
To get an approximation
to the integral.

632
00:37:37,670 --> 00:37:43,410
Which we know is the
area under the curve.

633
00:37:43,410 --> 00:37:49,730
So here's what the
Riemann sum is.

634
00:37:49,730 --> 00:37:51,020
It's the following thing.

635
00:37:51,020 --> 00:37:56,030
You take y_0 plus y_1
plus... up to y_(n-1).

636
00:37:56,030 --> 00:37:57,520
And you multiply by delta x.

637
00:37:57,520 --> 00:38:00,040
That's it.

638
00:38:00,040 --> 00:38:07,010
Now, this one is the
one with left endpoints.

639
00:38:07,010 --> 00:38:09,350
So the left-hand sum.

640
00:38:09,350 --> 00:38:11,120
There's also a right one.

641
00:38:11,120 --> 00:38:13,210
Which is, if you start
at the right-hand ends.

642
00:38:13,210 --> 00:38:17,830
And that will go
from y_1 to y_n.

643
00:38:17,830 --> 00:38:22,170
OK, so this one
is the right-hand.

644
00:38:22,170 --> 00:38:25,170
Right Riemann sum.

645
00:38:25,170 --> 00:38:33,110
Those are the two
that we did before.

646
00:38:33,110 --> 00:38:35,940
Now I'm going to describe
to you the next two.

647
00:38:35,940 --> 00:38:42,410
They have a similar
pattern to them.

648
00:38:42,410 --> 00:38:51,200
And the one with trapezoids
requires a picture.

649
00:38:51,200 --> 00:38:53,110
Here's a shape.

650
00:38:53,110 --> 00:39:00,690
And here's a bunch of values.

651
00:39:00,690 --> 00:39:08,870
And we're trying to estimate
the size of these chunks.

652
00:39:08,870 --> 00:39:10,950
And now, instead of doing
something stupid, which

653
00:39:10,950 --> 00:39:14,050
is to draw horizontal
lines in rectangles,

654
00:39:14,050 --> 00:39:16,320
we're going to do something
slightly more clever.

655
00:39:16,320 --> 00:39:21,550
Which is to draw straight
lines that are diagonal.

656
00:39:21,550 --> 00:39:23,177
You see that many
of them actually

657
00:39:23,177 --> 00:39:25,510
coincide probably pretty
closely with what I drew there.

658
00:39:25,510 --> 00:39:29,650
Although if they're curved,
they miss by a little bit.

659
00:39:29,650 --> 00:39:32,270
So this is called
the trapezoidal rule.

660
00:39:32,270 --> 00:39:35,660
Because if you pick one of
these shapes, say this is y_2

661
00:39:35,660 --> 00:39:40,120
and this is y_3, if you
pick one of these shapes,

662
00:39:40,120 --> 00:39:44,440
this height here is y_2,
and this height is y_3,

663
00:39:44,440 --> 00:39:47,720
and this base is delta x.

664
00:39:47,720 --> 00:39:52,590
This is a trapezoid.

665
00:39:52,590 --> 00:40:01,500
So this being a trapezoid,
I can figure out its area.

666
00:40:01,500 --> 00:40:02,760
And what do I get?

667
00:40:02,760 --> 00:40:12,020
I get the base times
the average height.

668
00:40:12,020 --> 00:40:14,470
If you think about
if, you work out

669
00:40:14,470 --> 00:40:16,060
what happens when
you do something

670
00:40:16,060 --> 00:40:18,220
with a straight line
on top, like that,

671
00:40:18,220 --> 00:40:21,080
you'll get this average.

672
00:40:21,080 --> 00:40:31,450
So this is the average
height of the trapezoid.

673
00:40:31,450 --> 00:40:34,580
And now I want to add up.

674
00:40:34,580 --> 00:40:41,600
I want to add them all
up to get my formula

675
00:40:41,600 --> 00:40:45,190
for the trapezoidal rule.

676
00:40:45,190 --> 00:40:46,090
So what do I do?

677
00:40:46,090 --> 00:40:50,210
I have delta x
times the first one.

678
00:40:50,210 --> 00:40:53,040
Which is (y_0 + y_1) / 2.

679
00:40:53,040 --> 00:40:55,010
That's the first trapezoid.

680
00:40:55,010 --> 00:40:59,520
The next one is (y_1 + y_2) / 2.

681
00:40:59,520 --> 00:41:01,710
And this keeps on going.

682
00:41:01,710 --> 00:41:07,790
And at the end, I have
(y_(n-2) + y_(n-1)) / 2.

683
00:41:07,790 --> 00:41:10,110
And then last of all, I
have (y_(n-1) + y_n) / 2.

684
00:41:14,610 --> 00:41:16,560
That's a very long formula here.

685
00:41:16,560 --> 00:41:21,390
We're going to simplify it
quite a bit in just a second.

686
00:41:21,390 --> 00:41:22,600
What's this equal to?

687
00:41:22,600 --> 00:41:28,820
Well, notice that I get
y_0 / 2 to start out with.

688
00:41:28,820 --> 00:41:32,930
And now, y_1 got
mentioned twice.

689
00:41:32,930 --> 00:41:35,400
Each time with a factor of 1/2.

690
00:41:35,400 --> 00:41:39,405
So we get a whole y_1 in here.

691
00:41:39,405 --> 00:41:40,780
And the same thing
is going to be

692
00:41:40,780 --> 00:41:43,940
true of all the middle terms.

693
00:41:43,940 --> 00:41:48,580
You're going to get y_2 and
all the way up to y_(n-1).

694
00:41:48,580 --> 00:41:53,750
But then, the last one is
unmatched. y_n is only 1/2,

695
00:41:53,750 --> 00:41:59,790
only counts 1/2.

696
00:41:59,790 --> 00:42:17,590
So here is what's known
as the trapezoidal rule.

697
00:42:17,590 --> 00:42:19,430
Now, I'd like to
compare it for you

698
00:42:19,430 --> 00:42:25,480
to the Riemann sums, which are
sitting just to the left here.

699
00:42:25,480 --> 00:42:28,900
Here's the left one, and
here's the right one.

700
00:42:28,900 --> 00:42:32,570
If you take the average
of the left and the right,

701
00:42:32,570 --> 00:42:34,960
that is, a half of this
plus a half of that,

702
00:42:34,960 --> 00:42:36,370
there's an overlap.

703
00:42:36,370 --> 00:42:39,250
The y_1 through y_n
things are listed in both.

704
00:42:39,250 --> 00:42:41,860
But the y_0 only
gets counted 1/2

705
00:42:41,860 --> 00:42:44,170
and the y_n only
gets counted 1/2.

706
00:42:44,170 --> 00:42:46,740
So what this is, is this
is the symmetric compromise

707
00:42:46,740 --> 00:42:49,710
between the two Riemann sums.

708
00:42:49,710 --> 00:42:54,410
This is actually equal
to the left Riemann

709
00:42:54,410 --> 00:43:02,020
sum plus the right
Riemann sum divided by 2.

710
00:43:02,020 --> 00:43:10,600
It's the average of them.

711
00:43:10,600 --> 00:43:13,340
Now, this would be great and
it does look like it's closer.

712
00:43:13,340 --> 00:43:16,500
But actually it's not as
impressive as it looks.

713
00:43:16,500 --> 00:43:18,910
If you actually
do it in practice,

714
00:43:18,910 --> 00:43:23,190
it's not very efficient.

715
00:43:23,190 --> 00:43:25,120
Although it's way better
than a Riemann sum,

716
00:43:25,120 --> 00:43:27,560
it's still not good enough.

717
00:43:27,560 --> 00:43:37,250
So now I need to describe
to you the fancier rule.

718
00:43:37,250 --> 00:43:52,010
Which is known as
Simpson's Rule.

719
00:43:52,010 --> 00:44:00,160
And so, this is, if
you like, 3, Method 3.

720
00:44:00,160 --> 00:44:09,770
The idea is again to
divide things into chunks.

721
00:44:09,770 --> 00:44:16,690
But now it always
needs n to be even.

722
00:44:16,690 --> 00:44:20,750
In other words, we're going
to deal not with just one box,

723
00:44:20,750 --> 00:44:24,800
we're going to deal
with pairs of boxes.

724
00:44:24,800 --> 00:44:28,370
Here's delta x, and
here's delta x again.

725
00:44:28,370 --> 00:44:34,720
And we're going to study
the area of this piece here.

726
00:44:34,720 --> 00:44:39,270
So let me focus
just on that part.

727
00:44:39,270 --> 00:44:50,700
Let's reproduce it over here.

728
00:44:50,700 --> 00:44:52,684
And here's the delta
x, here's delta x.

729
00:44:52,684 --> 00:44:54,350
And of course there
are various heights.

730
00:44:54,350 --> 00:44:57,020
This starts out at
y_0, this is y_2,

731
00:44:57,020 --> 00:45:03,490
and this middle segment is y_1.

732
00:45:03,490 --> 00:45:08,200
Now, the approximating curve
that we're going to use

733
00:45:08,200 --> 00:45:14,690
is a parabola.

734
00:45:14,690 --> 00:45:17,400
That is, we're going to fit a
parabola through these three

735
00:45:17,400 --> 00:45:25,240
points.

736
00:45:25,240 --> 00:45:28,500
And then we're going to use
that as the approximating area.

737
00:45:28,500 --> 00:45:32,360
Now, it doesn't look like--
This looks like it misses.

738
00:45:32,360 --> 00:45:34,262
But actually, most
functions mostly

739
00:45:34,262 --> 00:45:35,720
wiggle either one
way or the other.

740
00:45:35,720 --> 00:45:36,490
They don't switch.

741
00:45:36,490 --> 00:45:38,800
They don't have
inflection points.

742
00:45:38,800 --> 00:45:41,320
So, this is a lousy,
at this scale.

743
00:45:41,320 --> 00:45:42,890
But when we get to
a smaller scale,

744
00:45:42,890 --> 00:45:45,110
this becomes really fantastic.

745
00:45:45,110 --> 00:45:47,430
As an approximation.

746
00:45:47,430 --> 00:45:52,400
Now, I need to tell you
what the arithmetic is.

747
00:45:52,400 --> 00:45:56,980
And in order to save
time, it's on your problem

748
00:45:56,980 --> 00:46:00,080
set what the actual formula is.

749
00:46:00,080 --> 00:46:05,440
But I'm going to tell you
how to think about it.

750
00:46:05,440 --> 00:46:09,530
I want you to think
about it as follows.

751
00:46:09,530 --> 00:46:20,530
So the area under
the parabola is

752
00:46:20,530 --> 00:46:30,800
going to be a base times
some kind of average height.

753
00:46:30,800 --> 00:46:33,770
And the base here,
you can already see.

754
00:46:33,770 --> 00:46:36,020
It's 2 delta x.

755
00:46:36,020 --> 00:46:39,430
The base is 2 delta x.

756
00:46:39,430 --> 00:46:40,920
Now, the average
height is weird.

757
00:46:40,920 --> 00:46:43,070
You have to work out what
it is for a parabola,

758
00:46:43,070 --> 00:46:46,420
depending on those three
numbers, y_0, y_1, and y_2.

759
00:46:46,420 --> 00:46:48,760
And it turns out to be
the following formula.

760
00:46:48,760 --> 00:46:50,440
It has to be an
average, but it's

761
00:46:50,440 --> 00:46:51,910
an interesting weighted average.

762
00:46:51,910 --> 00:46:53,587
So this was the
punchline, if you like.

763
00:46:53,587 --> 00:46:56,170
Is that they are such things as
interesting weighted averages.

764
00:46:56,170 --> 00:46:58,680
This one's very simple, it
just involves three numbers.

765
00:46:58,680 --> 00:46:59,980
But it's still interesting.

766
00:46:59,980 --> 00:47:00,771
It's the following.

767
00:47:00,771 --> 00:47:07,260
It turns out to be
(y_0 + 4y_1 + y_2) / 6.

768
00:47:07,260 --> 00:47:08,160
Why divided by 6?

769
00:47:08,160 --> 00:47:10,730
Well, it's supposed
to be an average.

770
00:47:10,730 --> 00:47:13,640
So the total is 1 + 4
+ 1 of these things.

771
00:47:13,640 --> 00:47:16,090
And 6 is in the denominator.

772
00:47:16,090 --> 00:47:19,790
So it emphasizes the
middle more than the sides.

773
00:47:19,790 --> 00:47:23,440
And that's what happens
with a parabola.

774
00:47:23,440 --> 00:47:30,580
So this is a computation
which is on your homework.

775
00:47:30,580 --> 00:47:34,440
And now we can put this together
for the full Simpson's Rule

776
00:47:34,440 --> 00:47:36,190
formula.

777
00:47:36,190 --> 00:47:51,770
Which I'll put up over here.

778
00:47:51,770 --> 00:47:56,350
We have here 2 delta
x, and we divide by 6.

779
00:47:56,350 --> 00:48:03,470
And then we have (y_0 +
4y_1 + y_2) / 6 plus--

780
00:48:03,470 --> 00:48:05,390
That's the first chunk.

781
00:48:05,390 --> 00:48:09,130
Now, the second chunk, maybe
I'll just put it in here,

782
00:48:09,130 --> 00:48:11,820
starts-- This is x_2.

783
00:48:11,820 --> 00:48:14,050
This is x_0.

784
00:48:14,050 --> 00:48:15,620
And it goes all the way to x_4.

785
00:48:15,620 --> 00:48:18,800
So x_2, x_3, x_4.

786
00:48:18,800 --> 00:48:21,920
So the next one involves
the indices 2, 3 and 4.

787
00:48:21,920 --> 00:48:31,580
So this is (y_2 +
4y_3 + y_4) / 6.

788
00:48:31,580 --> 00:48:34,454
Oh, oh, oh, oh, no.

789
00:48:34,454 --> 00:48:35,870
I think I'll get
rid of these 6's.

790
00:48:35,870 --> 00:48:37,990
I have too many 6's.

791
00:48:37,990 --> 00:48:39,850
Alright.

792
00:48:39,850 --> 00:48:41,057
Let's get rid of them here.

793
00:48:41,057 --> 00:48:41,890
Let's take them out.

794
00:48:41,890 --> 00:48:44,040
Put them out here.

795
00:48:44,040 --> 00:48:47,870
Thank you.

796
00:48:47,870 --> 00:48:51,650
All the way to the end, which is
y_(n-2) plus 2y_(n-1)-- sorry,

797
00:48:51,650 --> 00:48:53,810
plus 4y_(n-1) plus y_n.

798
00:48:57,050 --> 00:49:01,580
I was about to divide
by 6, but you saved me.

799
00:49:01,580 --> 00:49:04,350
So here are all the chunks.

800
00:49:04,350 --> 00:49:06,960
Now, what does this
pattern come out to be?

801
00:49:06,960 --> 00:49:10,740
This comes out to
be the following.

802
00:49:10,740 --> 00:49:16,950
This is 1, 4, 1, added to
1, 4, 1 added to 1, 4, 1.

803
00:49:16,950 --> 00:49:17,760
You add them up.

804
00:49:17,760 --> 00:49:22,140
You get 1, 4, and then there's a
repeat, so you get a 2 and a 4,

805
00:49:22,140 --> 00:49:24,170
and a 2 and a 4 and a 1.

806
00:49:24,170 --> 00:49:29,250
So the pattern is that it starts
out with 1's on the far ends.

807
00:49:29,250 --> 00:49:30,910
And then 4's next in.

808
00:49:30,910 --> 00:49:35,230
And then it alternates
2's and 4's in between.

809
00:49:35,230 --> 00:49:45,440
So the full pattern of
Simpson's Rule is delta x / 3,

810
00:49:45,440 --> 00:49:48,720
I have now succeeded in
canceling this 2 with this 6

811
00:49:48,720 --> 00:49:50,740
and getting out
that factor of 2.

812
00:49:50,740 --> 00:50:00,966
And then here I have y0 +
4 y1 + 2 y2 + 4 y3 + ...

813
00:50:00,966 --> 00:50:03,520
It keeps on going and keeps
on going and keeps on going.

814
00:50:03,520 --> 00:50:10,820
And in the end it's 2 y_(n
- 2) + 4 y_(n - 1) + y_n.

815
00:50:10,820 --> 00:50:14,860
So again, 1 and a 4 to start.

816
00:50:14,860 --> 00:50:16,410
4 and a 1 to end.

817
00:50:16,410 --> 00:50:20,170
And then alternating 2's
and 4's in the middle.

818
00:50:20,170 --> 00:50:25,650
And this weird weighted
average is way better.

819
00:50:25,650 --> 00:50:28,890
As I will show you next time.