The Binomial Distributions 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) 2nd (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