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