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Binomial Distribution

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

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Binomial Distribution

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  1. Binomial Distribution

  2. Probability of Binary Events • Probability of success = p • p(success) = p • Probability of failure = q • p(failure) = q • p+q = 1 • q = 1-p

  3. 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.

  4. 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)

  5. Perm & Comb 3 • 3 flips • HHH, • HHT, HTH, THH • HTT, THT, TTH • TTT • All permutations equally likely = p*p*p = .53 = .125 = 1/8. • p(1 Head) = 3/8

  6. 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.

  7. 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

  8. Binomial 2

  9. Binomial 3 • Flip coins and compare observed to expected frequencies

  10. Binomial 4 • Find expected frequencies for number of 1s from a 6-sided die in five rolls.

  11. 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.

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