1. The Binomial Distributions
Statistics 101
The Binomial Distributions

2. 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

3. 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)

4. 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 So X has the binomial distribution B(5, 0.25)

5. 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.

6. 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.

7. 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)

8. Finding binomial probabilities
Probability histogram for the binomial distribution with n=10 and p=0.1

9. 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
(10, .1, 1) returns
Sum and we get
Or about 74% of all samples will contain no more than 1 bad switch

10. 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