2) There is a fixed number of observations n. 3) The n observations are all independent

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1 Chapter 8 Binomial and Geometric Distributions The binomial setting consists of the following 4 characteristics: 1) Each observation falls into one of two categories success or failure 2) There is a fixed number of observations n 3) The n observations are all independent 4) The probability of success, p, is the same for each observation The distribution of the number of successes in the binomial setting is the binomial distribution with parameters n and p. n represents the number of observations. p represents the probability of success on any one observation. The possible values of X are the whole numbers from 0 to n. As an abbreviation, we say B(n, p). 1

2 Do the following situations describe the binomial setting? If so, describe their distribution. 1) A child has 0.25 chance of getting type O blood if both of his parents are carriers for type O blood. Let X = the number of children among these parent s 5 children that have type O blood. 2) Deal 10 cards from a deck of cards and count the number of red cards. 3) Choose an SRS of 10 switches from a shipment of switches (of which 10% are bad) Let X = the # of bad switches in the SRS. Choose an SRS of 10 switches from a shipment of 10,000 switches (of which 10% are bad) Let X = the # of bad switches in the SRS. What is the probability that no more than 1 of the switches in the sample fail inspection? P( X 1) = P( X = 0) + P( X + 1) 2

3 The Formulas!!! Binomial Pdf is found by computing the formula: n k P( X = k) = p 1 p k ( ) n k Where n n! is equal to: k k ( n k )!! n k Represents the number of ways of arranging k successes among n observations. Ex: Find the probability of getting 5 correct on the multiple choice test we did yesterday. b) Now find the probability of getting 5 or fewer correct. 3

4 If a basketball player makes 25% of her free throws, what is the mean number you would expect her to make in 12 free throws? This leads us to the mean and standard deviation of a binomial random variable. µ = np σ = np(1 p) Ex: Find the mean and standard deviation of the number of bad switches is an SRS of 10 switches with the information we used before. 4

5 Using your calculator to compute binomial probabilities: Corrine is a basketball player who makes 75% of her free throws over the course of a season. In a key game she shoots 12 free throws and only makes 7 of them. Is it unusual for her to make 7 of 12 free throws if she is really a 75% free throw shooter? You can use your calculator to compute the binomial probability by entering binompdf(n, p, X). While the pdf command calculates one probability, the cdf command calculates the sum of the probabilities up to (and including) X. The command for cdf is binomcdf(n, p, X) Let s revisit the example with Corinne and her free throws. Corrine is a basketball player who makes 75% of her free throws over the course of a season. In a key game she shoots 12 free throws and only makes 7 of them. The fans think she choked because she was nervous. Is it unusual for Corinne to perform this poorly? 5

6 Just as with most distributions we discuss, we want to be able to use a normal distribution to describe a binomial setting. When n is large, we can use normal probability calculations to approximate binomial probabilities. How large is large? As a rule of thumb, we will use: np 10 and n(1 p) 10 So we will say when n is large, the distribution of X is approximately normal N ( np, np(1 p) ) Ex: Are attitudes towards shopping changing? Sample surveys show that fewer people are shopping than in the past. A recent survey asked a nationwide random sample of 2500 adults if they agreed or disagreed that I like buying new clothes, but shopping is often frustrating and time consuming. Suppose that in fact 60% of all US residents would say agree if asked the same question. What is the probability that 1520 or more of the sample agree? 6

7 Simulating Binomial Experiments You can use the randbin key on your calculator to simulate binomial situations. If we look at the example with our basketball player making 75% of her shots and shooting 12 shots, we can run a simulation to see how many shots out of 12 she would make on average or to see how many times she would make 7 or fewer shots. 7

Chapter 8.1.notebook. December 12, Jan 17 7:08 PM. Jan 17 7:10 PM. Jan 17 7:17 PM. Pop Quiz Results. Chapter 8 Section 8.1 Binomial Distribution

Chapter 8.1.notebook. December 12, Jan 17 7:08 PM. Jan 17 7:10 PM. Jan 17 7:17 PM. Pop Quiz Results. Chapter 8 Section 8.1 Binomial Distribution Chapter 8 Section 8.1 Binomial Distribution Target: The student will know what the 4 characteristics are of a binomial distribution and understand how to use them to identify a binomial setting. Process