Statistics 101 The Binomial Distributions The Binomial Setting

Statistics 101 The Binomial Distributions

The Binomial Setting • Each observation falls into one of two categories (success or failure) • Fixed n • The observations are all independent • The probability of success, p, is the same for each observation

Binomial Distribution • The distribution of the count X of successes in a binomial setting is the binomial distribution with parameters n and p. • n is the number of observations • p is the probability of success • X is B(n, p)

Examples • Blood type is inherited. If both parents carry genes for the O and A blood types, each child has a probability 0. 25 of getting two O genes. Different children inherit independently. The number of O blood types among 5 children of these parents is the count X of successes in 5 independent observations with probability 0. 25. So X has the binomial distribution B(5, 0. 25)

Example • Deal 10 cards from a shuffled deck and count the number X of red cards. Success is a red card. Is this a binomial distribution? • No, Because the observations are not independent therefore, it is not a binomial distribution.

Try exercise 8. 1 on pg 441. • (a) No: There is no fixed n (i. e. , there is no definite upper limit on the number of defects). • (b) Yes: It is reasonable to believe that all responses are independent (ignoring any “peer pressure”), and all have the same probability of saying “yes” since they are randomly chosen from the population. Also, a “large city” will have a population over 1000 (10 times as big as the sample). • (c) Yes: In a “Pick 3” game, Joe’s chance of winning the lottery is the same every week, so assuming that a year consists of 52 weeks (observations), this would be binomial.

Example 8. 5 Inspecting Switches A quality engineer selects an SRS of 10 switches from a large shipment for detailed inspection. Unknown to the engineer, 10% of the switches in the shipment fail to meet the specifications. What is the probability that no more than 1 of the 10 switches in the sample fail inspection? B(10, 0. 1)

Finding binomial probabilities Probability histogram for the binomial distribution with n=10 and p=0. 1

Calculations • We want to calculate P(X< 1) = P(x=0) + P(x = 1) • TI-83 command binompdf(n, p, X) • 2 nd (Distri)/0: binompdf • (10, . 1, 0) returns. 3486784401 • (10, . 1, 1) returns 0. 387420489 • Sum and we get 0. 7361 • Or about 74% of all samples will contain no more than 1 bad switch

Cumulative distribution function (cdf) • Calculates the sum of the probabilities for 0, 1, 2, up to the value of X. • For the count X of defective switches previously done • binomcdf (10, . 1, 1) returns 0. 736098903