Binomial Distribution Probability of Binary Events Probability of

Binomial Distribution

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

Permutations & Combinations 1 • • • Suppose we flip a coin 2 times HH HT TH TT 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. • p(1 Head) = 3/8

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 2 5 . 03125 4 . 15625 3 . 3125 2 . 3125 1 . 15625 0 . 03125

Binomial 3 • Flip coins and compare observed to expected frequencies

Binomial 4 • Find expected frequencies for number of 1 s 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.