The Practice of Statistics Third Edition Chapter 8: The Binomial and Geometric Distributions Copyright © 2008 by W. H. Freeman & Company Daniel S. Yates

Ex. The number of 3’s rolled by a single die in ten trials is a binomial random variable n=10p=1/6 symbol X -> B(10,1/6) Ex. The number of blue tiles chosen from a box containing 8 blue tiles 2 red tiles and 10 yellow tiles in 5 trials. with replacement - n=5, p=0.4 Symbol X-> B(5,0.4) without replacement – Not binomial; p changes and observations are not independent

TI-83,84 calculator There are two calculator functions that are of great help in calculating binomial probabilities. binompdf(n,p,x) - gives the probability of each value of x that you input. binomcdf(n,p,x) - calculates the cumulative probability for: P(x< # of successes) = P(0) + P(1) … …+ P(# of successes)

Ex. What is the probability that a couple planning to have 8 children will have 1)exactly 5 boys. 2) at most 5 boys 1) P=0.5, n=8, X=5 symbol X-> B(8,0.5) Calculator – P(x=5) = binompdf(8,0.5,5) = ) P=0.5, n=8, X=5 symbol X-> B(8,0.5) Calculator – P(x<5) = binomcdf(8,0.5,5) =

Binomial Formula Calculate the probability of 3 sixes in 10 rolls of a die. P(success) = 1/6; P(failure) = 5/6 -One possibility; but there are many ways to get 3 sixes out of ten rolls. Obs outcome6No six ns 66 probability1/65/6 1/6 5/6 The probability for each of the many ways is equal to (1/6) 3 x (5/6) 7

The binomial coefficient gives the number of ways this can happen; = nCr on calculator

Ex. What is the probability of having 5 boys out of 7 children? P(x=5) = 7 C 5 (0.5) 5 (0.5) 2 = TI-83,84 does it for you – binompdf(7,0.5,5)=0.164

Ex. What is the probability that you roll your first 5 on your fifth roll of a die? “Success” -> roll a 5 P(success) = 1/6 “Failure” -> roll anything else P(failure) = 5/6 Roll1 st 2 nd 3 rd 4 th 5th Probability5/6 1/6 P(x=5) = (5/6) 4 x(1/6) 1 = 0.08

What is the probability that it takes at most 5 rolls of the die to get a 5? P(x=1) + P(x=2) + P(x=3) + P(x=4) + P(x=5) (1/6) + (5/6 x 1/6) +( (5/6) 2 x 1/6) + ((5/6) 3 x 1/6) +( (5/6) 4 x 1/6) = TI-83,84 geometpdf(p,x) and geometcdf(p,x) P(x=n)P(x<n)