1. More Discussion of the Binomial Distribution: Comments & Examplesj
Thermo & Stat Mech - Spring Class 16

2. Requirements justifying use of the Binomial Distribution:
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.
Requirements justifying use of
the Binomial Distribution:
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.

3. Binomial Distribution
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Binomial Distribution
The Probability of n Successes
out of N Attempts is:
p = Probability of a Success
q = Probability of a Failure
q = 1 – p, (p + q)N = 1

4. Mean of the Binomial Distribution
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Mean of the Binomial Distribution l
Thermo & Stat Mech - Spring Class 16

5. Standard Deviation (s) of the Binomial Distribution
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Standard Deviation (s) of the Binomial Distribution 2 1  3

6. For The Binomial Distribution
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For The Binomial Distribution

7. Common Notation for the Binomial Distribution
r items of one type & (n – r) of a second type
can be arranged in nCr ways. Here: ≡
nCr is called the binomial coefficient
In this notation, the probability distribution is:
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 .

8. Binomial Distribution: Example
Problem: 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.
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?)

9. Binomial Distribution, n = 11 Number of Defective Bulbs, r
Probability
11Crpr(1-p)n-r
p = 0.04
11C0 (0.04)0(0.96)11 = 1
11C1 (0.04)1(0.96)10 = 2
11C2 (0.04)2(0.96)9 = 3
11C3 (0.04)3(0.96)8 = l
Thermo & Stat Mech - Spring Class 16

10. P(r = 3 defective bulbs) =
Question 1: Probability of finding exactly
three defective bulbs in a sample?
P(r = 3 defective bulbs) =
W11(r = 3) = l
Thermo & Stat Mech - Spring Class 16

11. P(r = 3 defective bulbs) =
Question 1: Probability of finding exactly
three defective bulbs in a sample?
P(r = 3 defective bulbs) =
W11(r = 3) =
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 – – = l
Thermo & Stat Mech - Spring Class 16

12. Binomial Distribution, Same Problem, Larger r
Number of Defective Bulbs, r
Probability
11Crpr(1-p)n-r
11C0(0.04)0(0.96)11 = 1
11C1 (0.04)1(0.96)10 = 2
11C2 (0.04)2(0.96)9 = 3
11C3 (0.04)3(0.96)8 = 4
11C4 (0.04)4(0.96)7 = 5
11C5 (0.04)5(0.96)6 =
Thermo & Stat Mech - Spring Class 16 l

13. Thermo & Stat Mech - Spring 2006 Class 16
Binomial
Distribution
n = 11, p = 0.04
Distribution of
Defective Items
Distribution of
Good Items l
Thermo & Stat Mech - Spring Class 16

14. The Coin Flipping Problem
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The Coin Flipping Problem
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: l
Thermo & Stat Mech - Spring Class 16

15. Binomial Distribution for Flipping 1000 Coins
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Binomial Distribution for Flipping 1000 Coins
Note: The distribution peaks around n = 500
successes (heads), as we would expect
n = Np = 500 l
Thermo & Stat Mech - Spring Class 16

16. Binomial Distribution for Selected Values of n & p
Thermo & Stat Mech - Spring Class 16

17. Binomial Distribution for Selected Values of n & p
n = 5, p = n = 5, p = 0.5
n = 10, p = 0.5 l
Thermo & Stat Mech - Spring Class 16

18. Binomial Distribution for Selected Values of n & p
Thermo & Stat Mech - Spring Class 16

19. Binomial Distribution for Selected Values of n & p