thermo & stat mech - spring 2006 class 16 more discussion of the binomial distribution: comments...
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Thermo & Stat Mech - Spring 2006 Class 16
More Discussion of the Binomial Distribution: Comments & Examples
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The Binomial Distribution applies ONLY to cases where there are only 2 possible outcomes: heads or tails, success or failure, defective or good item, etc.
The requirements justifying the use of the Binomial Distribution are:
1. The experiment must consist of n identical trials.2. Each trial must result in only one of 2 possible
outcomes. 3. The outcomes of the trials must be statistically
independent.4. All trials must have the same probability for a
particular outcome.
Binomial DistributionThe Probability of n Successes
out of N Attempts is:
p = Probability of a Successq = Probability of a Failure
q = 1 – p(p + q)N = 1
nNnqpnNn
NnP
)!(!!
)(
Thermo & Stat Mech - Spring 2006 Class 16
Mean of the Binomial Distribution
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Standard Deviation () of the Binomial Distribution
pNqpNppNpNn
qpNpNqpNpn
qppNp
pqpp
pp
pn
nPp
pnnPn
NN
NN
nn
1
))(1()(
)()(
)()(
2
212
12
222
Npq
NpqpNpNNpq
pNpNqpN
nn
222
22
222
)()(
)(
1 2
3
For The Binomial Distribution
Npq
n
Npq
pNn
Common Notation for the Binomial Distributionr items of one type & (n – r) of a second type can be arranged in nCr ways. Here:
r)!(nr!
n!Crn
nCr is called the binomial coefficientIn this notation, the probability distribution can be written:
Wn(r) = nCrpr(1-p)n-r
≡ probability of finding r items of one type & n – r items of the other type. p = probability of a given item being of one type .
≡
Binomial Distribution: ExampleProblem: A sample of n = 11 electric bulbs is drawn every day from those manufactured at a plant. The probabilities of getting defective bulbs are random and independent of previous results. The probability that a given bulb is defective is p = 0.04.
1. What is the probability of finding exactly three defective bulbs in a sample?
(Probability that r = 3?)2. What is the probability of finding three or more defective bulbs in a sample?
(Probability that r ≥ 3?)
Thermo & Stat Mech - Spring 2006 Class 16
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Binomial Distribution, n = 11
Number of Defective Bulbs, r
Probability
11Crpr(1-p)n-r
p = 0.04
0 11C0 (0.04)0(0.96)11 = 0.6382
1 11C1 (0.04)1(0.96)10 = 0.2925
2 11C2 (0.04)2(0.96)9 = 0.0609
3 11C3 (0.04)3(0.96)8 = 0.0076
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Thermo & Stat Mech - Spring 2006 Class 16
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Question 1: Probability of finding exactly three defective bulbs in a sample?
P(r = 3 defective bulbs) = W11(r = 3) = 0.0076
Question 2: Probability of finding three or more defective bulbs in a sample?
P(r ≥ 3 defective bulbs) =
1- W11(r = 0) – W11(r = 1) – W11(r = 2) =
1 – 0.6382 - 0.2925 – 0.0609 = 0.0084l
Thermo & Stat Mech - Spring 2006 Class 16
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Binomial Distribution, Same Problem, Larger r
Number of Defective Bulbs, r
Probability
11Crpr(1-p)n-r
0 11C0(0.04)0(0.96)11 = 0.638239
1 11C1 (0.04)1(0.96)10 = 0.292526
2 11C2 (0.04)2(0.96)9 = 0.060943
3 11C3 (0.04)3(0.96)8 = 0.007618
4 11C4 (0.04)4(0.96)7 = 0.000635
5 11C5 (0.04)5(0.96)6 = 0.000037
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Thermo & Stat Mech - Spring 2006 Class 16
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BinomialDistribution
n = 11, p = 0.04
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Distribution ofDefective Items
Distribution ofGood Items
Thermo & Stat Mech - Spring 2006 Class 16
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• Consider a perfect coin. There are only 2 sides, so the probability associated with coin flipping is
The Binomial Distribution.• Problem: 6 perfect coins are flipped. What is the
probability that they land with n heads & 1 – n tails? Of course, this only makes sense if 0 ≤ n ≤ 6! For this case, the Binomial Distribution has the form:
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The Coin Flipping Problem
Thermo & Stat Mech - Spring 2006 Class 16
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Binomial Distribution for Flipping 1000 Coins
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Note: The distribution peaks around n = 500 successes (heads), as we would expect ( = 500)
Thermo & Stat Mech - Spring 2006 Class 16
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Binomial Distribution for Selected Values of n & p
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n = 20, p = 0.5n = 10, p = 0.1 &
n =10, p = 0.9
Thermo & Stat Mech - Spring 2006 Class 16
Binomial Distribution for Selected Values of n & p
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.0
.2
.4
.6
0 1 2 3 4 5
X
P(X)
.0
.2
.4
.6
0 1 2 3 4 5
X
P(X)
n = 5, p = 0.1 n = 5, p = 0.5
.0
.2
.4
.6
0 1 2 3 4 5
X
P(X)
.0
.2
.4
.6
0 1 2 3 4 5
X
P(X)
n = 10, p = 0.5
Thermo & Stat Mech - Spring 2006 Class 16
Binomial Distribution for Selected Values of n & p
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n = 5, p = 0.5
n = 20, p = 0.5
n = 100, p = 0.5
Binomial Distribution for Selected Values of n & p