8.1 The Binomial Distributions AP Statistics 8.1 The Binomial Distributions

Learning Objective: Determine if a binomial setting is valid Calculate the probability of a binomial distribution Calculate the mean and standard deviation of a binomial setting

Properties of the Binomial Distribution   1- success or failure 2- they’re independent 3- probability remains the same 4- there are a fixed number of observations

Which of the following represents a binomial distribution and why? Ex 1- Blood type is inherited. Each child born to a particular set of parents has a probability of 0.25 of having blood type O. Different children inherit independently of each other. Let X represent the number of children with O blood type in 5 independent observations.

So X has the binomial distribution with n= 5 and p= 0.25 Therefore, we say that X is B( 5 , 0.25 ).

Ex 2- Deal 10 cards from a shuffled deck and count the number X of red cards. There are 10 observations, and each gives either a red or a black card. A “success” is a red card. But the observations are not independent. If the first card is black, the second is more likely to be red because there are more red cards than black cards left in the deck.

For the following examples, X is a count For the following examples, X is a count. Does X have a binomial distribution? You observe the sex of the next 20 children born at a local hospital; X is the number of girls among them. A couple decides to continue to have children until their first girl is born. X is the total number of children the couple has. Joe buys a state lottery ticket every week. The count X is the number of times a year that he wins a prize.

For the following examples, X is a count For the following examples, X is a count. Does X have a binomial distribution? A student studies mathematics using computer-assisted instruction. After the lesson, the computer presents 10 problems. The student solves each problem and enters her answer. The computer gives additional instruction between problems if the answer is wrong. The count X is the number of problems the student gets right.

Exactly 2 kids have Type O Blood?

Binomial Formulas   Binomial Coefficient Binomial Probability

B(5, .25) Ex: What is the probability that no more than 2 of them have blood type O?

Using the calculator: Binompdf(n,p,k) Probability Distribution Function (PDF)- probability of exactly 1 event Cumulative Distribution Function (CDF)- probability of more than 1 event

Finding Binomial Probabilities Ex: An SRS of 10 switches is taken from a large shipment. 10% of the switches fail to meet specifications. What is the probability that no more than 1 of the 10 switches in the sample fails inspection?

What is the probability that more than 4 switches in the sample fail?

Mean of a Random Variable µ= np Standard Deviation of a Binomial Random Variable σ= √(np(1-p))

Ex: What is the mean and standard deviation of the binomial distribution for the bad switches. Recall n=10 and p=0.1? µ= np= (10)(0.1)= 1 σ= √((10)(0.1)(0.9))= √(0.9)= 0.9487