Binomial Distributions Section 4.2 Binomial Distributions

Binomial Experiments 4 Conditions for a Binomial Experiment Also: There are a fixed number of trials. (n) Each trial is independent. Each trial has 2 outcomes (p = Success or q = Failure.) The probability of success for each trial is the same. Also: p + q = 1 The random variable x is a count of the number of successes in n trials. The central problem is to find the probability of x successes out of n trials. Where x = 0 or 1 or 2 … n. Examples: roll a die eight times and count the number of times a 4 appeared. Throw 12 basketball free shots and count the number of times the ball went through the basket. Make 15 sales calls and count the number of sales made.

Is it a binomial experiment? An experiment in which a basketball player who historically makes 80% of his free throws is asked to shoot 3 free throws and the number of made free throws is recorded. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____ q = _____ n = _____ x = _______________

Is it a binomial experiment? 2. The number of people with blood type O-negative based upon a simple random sample of size 10 is recorded. According to the Information Please Almanac, 6% of the human population is blood type 0-negative. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____ q = _____ n = _____ x = _______________

Is it a binomial experiment? 3. A probability experiment in which three cards are drawn from a deck without replacement and the number of aces is recorded. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____ q = _____ n = _____ x = _______________

Is it a binomial experiment? 4. A random sample of 15 college seniors is conducted, and the individuals selected are asked to state their ages. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____ q = _____ n = _____ x = _______________

Is it a binomial experiment? 5. An experimental drug is administered to 100 randomly selected individuals, with the number of individuals responding favorably recorded. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____ q = _____ n = _____ x = _______________

Is it a binomial experiment? A poll of 1200 registered voters is conducted in which the respondents are asked whether they believe Congress shoulc reform Social Security. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____ q = _____ n = _____ x = _______________

Is it a binomial experiment? 7. A baseball player who reaches base safely 30% of the time is allowed to bat until he reaches base safely for the third time. The number of at-bats required is recorded. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____ q = _____ n = _____ x = _______________

Is it a binomial experiment? 8. An investor randomly purchases 10 stocks listed on the New York Stock Exchange. Historically, the probability that a stock listed on the NYSE will increase in value over the course of a year is 48%. The number of stocks that increase in value is recorded. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____ q = _____ n = _____ x = _______________

How do you construct Binomial Probability Distributions? If you have three children, what is the probability that you will have none, one, two, or three boys? Old Method: Construct a tree diagram. Compute probabilities. New Method: Find x, n, p, and q. Complete binomial distribution table using formula Have students take this quiz explaining that they should simply guess each answer.

Constructing Binomial Distributions If you have three children, what is the probability that you will have none, one, two, or three boys? X = _____ p = _____ q = _____ n = _____ X Number of boys P(x) = 1 2 3

Constructing Binomial Distributions Example 2 - #9 You are taking a multiple-choice quiz that consists of five questions. Each question has four possible answers, only one of which is correct. To complete the quiz, you randomly guess the answer to each question. X = _____ p = _____ q = _____ n = _____ X P(x) =

Constructing Binomial Distributions Example 2 - #9 cont. Find the probability of the following: Exactly 3 answers correctly At least 3 answers correctly Less than 3 answers correctly X P(x) Xx

How do you find the Mean and Standard Deviation? If you have three children, what is the probability that you will have none, one, two, or three boys? X = 0,1,2,3 p = .5 q = .5 n = 3 x P(x) nCxpxq(n-x) xP(x) (x-)2P(x) 1 2 3

Page 184, #17 x P(x) xP(x) (x-μ)2P(x)

Homework Page 184 #5-8, 16, 18, 20