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# Binomial Distribution PowerPoint PPT Presentation

Binomial Distribution. Probability of Binary Events. Probability of success = p p(success) = p Probability of failure = q p(failure) = q p+q = 1 q = 1-p. Permutations & Combinations 1. Suppose we flip a coin 2 times H H H T T H T T

Binomial Distribution

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## Binomial Distribution

### Probability of Binary Events

• Probability of success = p

• p(success) = p

• Probability of failure = q

• p(failure) = q

• p+q = 1

• q = 1-p

### Permutations & Combinations 1

• Suppose we flip a coin 2 times

• H H

• H T

• T H

• T T

• Sample space shows 4 possible outcomes or sequences. Each sequence is a permutation. Order matters.

• There are 2 ways to get a total of one heads (HT and TH). These are combinations. Order does NOT matter.

### Perm & Comb 2

• HH, HT, TH, TT

• Suppose our interest is Heads. If the coin is fair, p(Heads) = .5; q = 1-p = .5.

• The probability of any permutation for 2 trials is ¼ = p*p, or p*q, or q*p, or q*q. All permutations are equally probable.

• The probability of 1 head in any order is 2/4 = .5 = HT+TH/(HH+HT+TH+TT)

### Perm & Comb 3

• 3 flips

• HHH,

• HHT, HTH, THH

• HTT, THT, TTH

• TTT

• All permutations equally likely = p*p*p = .53 = .125 = 1/8.

### Perm & Comb 4

• Factorials: N!

• 4! = 4*3*2*1

• 3! = 3*2*1

• Combinations: NCr

• The number of ways of selecting r combinations of N objects, regardless of order. Say 2 heads from 5 trials.

### Binomial Distribution 1

• Is a binomial distribution with parameters N and p. N is the number of trials, p is the probability of success.

• Suppose we flip a fair coin 5 times; p = q = .5

### Binomial 3

• Flip coins and compare observed to expected frequencies

### Binomial 4

• Find expected frequencies for number of 1s from a 6-sided die in five rolls.

### Binomial 5

• When p is .5, as N increases, the binomial approximates the Normal.

Probability for numbers of heads observed in 10 flips of a fair coin.