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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

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Probability of binary events
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
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
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
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
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
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
Binomial 3

  • Flip coins and compare observed to expected frequencies


Binomial 4
Binomial 4

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


Binomial 5
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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