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Binomial Distribution

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Binomial Distribution

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

- 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.

- 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)

- 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

- 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.

- 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

- Flip coins and compare observed to expected frequencies

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

- 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.