Binomial Distribution

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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 &amp; Combinations 1. Suppose we flip a coin 2 times H H H T T H T T

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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.