Random Variables Lesson 5 3 Binomial Random Variables

Random Variables Lesson 5. 3 Binomial Random Variables Statistics and Probability with Applications, 3 rd Edition Starnes & Tabor Bedford Freeman Worth Publishers

Binomial Random Variables Learning Targets After this lesson, you should be able to: ü Determine whether or not a given scenario is a binomial setting. ü Calculate probabilities involving a single value of a binomial random variable. ü Make a histogram to display a binomial distribution and describe its shape. Statistics and Probability with Applications, 3 rd Edition 2

The Binomial Distribution Overview • Imagine that a trial is repeated n times • Examples – A coin is tossed 5 times – A die is rolled 25 times – 50 chicken eggs are examined • Assume p (the probability of success) remains constant from trial to trial and that the trials are statistically independent of each other Statistics and Probability with Applications, 3 rd Edition 3

The Binomial Distribution When the same chance process is repeated several times, we are often interested in how many times a particular outcome occurs that is, • What is the probability of obtaining x successes in n trials? • Example – What is the probability of obtaining 2 heads from a coin that was tossed 5 times? – This type of scenario is known as a Bernoulli Trial Statistics and Probability with Applications, 3 rd Edition 4

Bernoulli Trials We have Bernoulli trials if: – there are two possible outcomes (success and failure). – the probability of success, p, is constant. – the trials are independent. Statistics and Probability with Applications, 3 rd Edition 5

The Binomial Model • A binomial setting arises when we perform several independent trials of the same chance process and record the number of times that a particular outcome (called a “success”) occurs. • A Binomial model tells us the probability for a random variable that counts the number of successes in a fixed number of Bernoulli trials. • Two parameters define the Binomial model: n, the number of trials; and, p, the probability of success. We denote this Binom(n, p). Statistics and Probability with Applications, 3 rd Edition 6

Binomial Random Variables When the same chance process is repeated several times, we are often interested in how many times a particular outcome occurs. Binomial Setting A binomial setting arises when we perform several independent trials of the same chance process and record the number of times that a particular outcome (called a “success”) occurs. The four conditions for a binomial setting are: • Binary? The possible outcomes of each trial can be classified as “success” or “failure. ” • Independent? Trials must be independent; that is, knowing the result of one trial must not tell us anything about the result of any other trial. • Number? The number of trials n of the chance process must be fixed in advance. • Success? There is the same probability p of success on each trial. Statistics and Probability with Applications, 3 rd Edition 7

Are these binomial settings? 1) Toss a coin 10 times and count the number of heads Yes 2) Deal 10 cards from a shuffled deck and count the number of red cards No, probability does not remain constant 3) Two parents with genes for O and A blood types are starting a family. Count the number of children with blood type O No, no fixed number Statistics and Probability with Applications, 3 rd Edition 8

Identifying binomial settings Determine whether or not the given scenario describes a binomial setting. Justify your answer. • (a) An American roulette wheel has 38 equally-sized slots into which a ball will fall randomly. There are 18 red slots, 18 black slots, and 2 green slots. Successive games of roulette will be played until the first time the ball falls into a green slot. • (b) Professional football teams in the NFL must play 16 regular season games. Each game begins with a coin toss to determine who gains possession of the ball first. Suppose one team is chosen and you record the number of times they win the coin toss in one season. Statistics and Probability with Applications, 3 rd Edition 9

Binomial Random Variables When checking the Binary condition, note that there can be more than two possible events per trial, however only one is defined as success and all others are defined as failure. In this way we are still really only talking about the two outcomes: success and failure. For Example: In the Pop Quiz, each question (trial) could have five possible answer choices: A, B, C, D, or E. If we define “success” as guessing the correct answer to a question, then “failure” occurs when the student guesses any of the four incorrect answer choices. Binomial Random Variable The count of successes X in a binomial setting is a binomial random variable. The possible values of X are 0, 1, 2, . . . , n. Statistics and Probability with Applications, 3 rd Edition 10

Binomial Random Variables How can we calculate probabilities involving binomial random variables? Binomial Probability Formula Suppose that X is a binomial random variable with n trials and probability p of success on each trial. The probability of getting exactly k successes in n trials (k = 0, 1, 2, . . . , n) is where The binomial probability formula looks complicated, but it is fairly straight forward if you know what each part of the formula represents: P(X = k) = (# of ways to get k successes in n trials) times (success probability) k (failure probability)n−k Statistics and Probability with Applications, 3 rd Edition 11

Using the binomial probability formula Professional football teams in the NFL must play 16 regular season games. Each game begins with a coin toss to determine who gains possession of the ball first. What is the probability that one team wins the coin toss in exactly 13 of their 16 games? Statistics and Probability with Applications, 3 rd Edition 12

BINOMIAL PROBABILITIES Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads? Statistics and Probability with Applications, 3 rd Edition 13

Binomial Probabilities on the Calculator • Binomialpdf(n, p, x) – this calculates the probability of a single binomial P(x = k) • Binomialcdf(n, p, x) – this calculates the cumulative probabilities from P(0) to P(k) (calculates cumulative probability from left to right only) Statistics and Probability with Applications, 3 rd Edition 14

The number of inaccurate gauges in a group of four is a binomial random variable. If the probability of a defect is 0. 1, what is the probability that only 1 is defective? Statistics and Probability with Applications, 3 rd Edition 15

Binomial Random Variables What does the probability distribution of a binomial random variable look like? Binomial Distribution The probability distribution of a binomial random variable is a binomial distribution. Any binomial distribution is completely specified by two numbers: the number of trials n of the chance process and the probability p of success on any trial. X = the number of children with type O blood This binomial distribution with n = 5 and p = 0. 25 has a clear right-skewed shape. Value Probability 0 1 2 3 4 5 0. 23730 0. 39551 0. 26367 0. 08789 0. 01465 0. 00098 Statistics and Probability with Applications, 3 rd Edition 16

LESSON APP 5. 3 Is the train binomial? According to New Jersey Transit, the 8: 00 A. M. weekday train from Princeton to New York City has a 90% chance of arriving on time on a randomly selected day. Suppose this claim is true. Choose 6 days at random. Let X = the number of days on which the train arrives on time. 1. Explain why this is a binomial setting. 2. Calculate and interpret P(X = 4). Statistics and Probability with Applications, 3 rd Edition 17

Binomial Random Variables Learning Targets After this lesson, you should be able to: ü Determine whether or not a given scenario is a binomial setting. ü Calculate probabilities involving a single value of a binomial random variable. ü Make a histogram to display a binomial distribution and describe its shape. Statistics and Probability with Applications, 3 rd Edition 18