5 Random Variables Lesson 5 4 Analyzing Binomial

5 Random Variables Lesson 5. 4 Analyzing Binomial Random Variables Statistics and Probability with Applications, 3 rd Edition Starnes & Tabor Bedford Freeman Worth Publishers

Analyzing Binomial Random Variables Learning Targets After this lesson, you should be able to: ü Calculate and interpret the mean and standard deviation of a binomial distribution. ü Find probabilities involving several values of a binomial random variable. ü Use technology to calculate cumulative binomial probabilities. Statistics and Probability with Applications, 3 rd Edition 2

Analyzing Binomial Random Variables To calculate the mean of a discrete random variable X, we use the formula µX = E(X) = x 1 p 1 + x 2 p 2 + x 3 p 3 +. . . Calculating the Mean of a Binomial Random Variable If a count X of successes has a binomial distribution with number of trials n and probability of success p, the mean of X is µX = np Statistics and Probability with Applications, 3 rd Edition 3

Analyzing Binomial Random Variables To calculate the standard deviation of a discrete random variable X, we use the formula Calculating the Standard Deviation of a Binomial Random Variable If a count X of successes has a binomial distribution with number of trials n and probability of success p, the standard deviation of X is Caution! Remember that these formulas for the mean and standard deviation work only for binomial distributions. Statistics and Probability with Applications, 3 rd Edition 4

Analyzing Binomial Random Variables According to the science of genetics, the genes children receive from their parents are independent from one child to another. Each child of a particular set of parents has probability 0. 25 of having type O blood. Suppose these parents have 5 children. Let X = the number of children with type O blood. xi 0 1 2 3 4 5 pi 0. 2373 0. 3955 0. 2637 0. 0879 0. 0147 0. 00098 Mean: Based on our formula for the mean: Standard Deviation: Based on our formula for the standard deviation Statistics and Probability with Applications, 3 rd Edition 5

Do you have loops, whorls, or arches? Mean and SD of a binomial distribution PROBLEM: The fingerprint of every finger can be classified into one of three basic patterns: a loop, a whorl, or an arch. Forensic experts have determined that about 65% of all fingers have loops, 30% have whorls, and 5% have arches. Suppose that these percentages are exactly correct and patterns are independent from one finger to the next. Select a person at random and classify each of their fingerprints on both hands. Let X = the number of fingerprints with a loop. (a) Calculate and interpret the mean of X. (b) Calculate and interpret the standard deviation of X. SOLUTION: The random variable X has a binomial distribution with n = 10 and p = 0. 65. • (a) If many people were selected, we’d expect 6. 5 fingerprints with a loop, on average, on an individual’s two hands. • (b) If many people were selected, the number of fingerprints with a loop on an individual’s two hands would typically vary from the mean of 6. 5 fingerprints by about 1. 508 fingerprints. Statistics and Probability with Applications, 3 rd Edition 6

Analyzing Binomial Random Variables According to the science of genetics, the genes children receive from their parents are independent from one child to another. Each child of a particular set of parents has probability 0. 25 of having type O blood. Suppose these parents have 5 children. Let X = the number of children with type O blood. xi 0 1 2 3 4 5 pi 0. 2373 0. 3955 0. 2637 0. 0879 0. 0147 0. 00098 What’s the probability that at most 1 of the children has type O blood? In symbols, it’s P(X ≤ 1). We can compute this cumulative binomial probability using the fact that P(X ≤ 1) = P(X=0) + P(X=1) = 0. 23730 + 0. 39551 = 0. 63281 What if we want to find the probability that at least 2 of the couple’s 5 children have type O blood? In symbols, that’s P(X ≥ 2). We could compute this probability using the fact that P(X ≥ 2) = P(X=2) + P(X=3) + P(X=4) + P(X=5) = 0. 26367 + 0. 08789 + 0. 01465 + 0. 00098 = 0. 36719 Statistics and Probability with Applications, 3 rd Edition 7

Feeling loop-y? Finding cumulative binomial probabilities PROBLEM: The fingerprint of every finger can be classified into one of three basic patterns: a loop, a whorl, or an arch. Forensic experts have determined that about 65% of all fingers have loops, 30% have whorls, and 5% have arches. Suppose that these percentages are exactly correct and patterns are independent from one finger to the next. Select a person at random and classify each of their fingerprints on both hands. Let X = the number of fingerprints with a loop. The probability distribution of X is shown below, with probabilities rounded to the nearest thousandth. Number of loops 0 1 2 3 4 5 6 7 8 9 10 0. 001 0. 004 0. 021 0. 069 0. 154 0. 238 0. 252 0. 176 0. 072 0. 013 Probability Victor has 3 loop patterns in his fingerprints and wonders what the probability is to have more than this. Find the probability that a randomly chosen person has 3 or fewer loop patterns. SOLUTION: P(X 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0. 000 + 0. 001 + 0. 004 + 0. 021 = 0. 026 Statistics and Probability with Applications, 3 rd Edition 8

Can Joy smell Parkinson’s disease? Cumulative binomial probabilities with technology PROBLEM: Joy Milne of the United Kingdom claimed that 6 years before her husband was diagnosed with Parkinson’s disease, she smelled a musky odor on him. As she spent more time around other Parkinson’s patients, she noticed a similar smell. Joy claimed that she could smell Parkinson’s disease. An experiment was devised to investigate her claim: 12 people were given identical shirts to wear for one day. Some of the patients had Parkinson’s and some did not. The shirts were then given to Joy one at a time and she had to decide whether or not each shirt was worn by a Parkinson’s patient or not. If Joy had no special ability to smell Parkinson’s, then she would just be guessing for each shirt. Let X = the number of correct decisions by someone who was just guessing. Joy made correct decisions for 11 of the 12 shirts. Are you convinced that Joy can smell Parkinson’s disease? Compute P(X 11) with technology to support your answer. SOLUTION: X is a binomial random variable with n = 12 and p = 1/2. P(X ≥ 11) = 1 – P(X ≤ 10) = 1 – 0. 9968 = 0. 0032 Joy had about a 3 in 1000 chance to make 11 or more correct decisions just by guessing. This is convincing evidence that Joy could smell Parkinson’s disease. Note: Joy’s only “mistake” actually turned out not to be a mistake. The person wearing that shirt was diagnosed with Parkinson’s 8 months after the experiment and Joy had insisted that he had Parkinson’s disease. Thus, Joy actually made 12 out of 12 correct decisions. (The probability to make that many correct decisions by guessing is about 0. 0003. ) Statistics and Probability with Applications, 3 rd Edition 9

LESSON APP 5. 4 Free lunch? A local fast-food restaurant is running a “Draw a three, get it free” lunch promotion. After each customer orders, a touch-screen display shows the message, “Press here to win a free lunch. ” A computer program then simulates one card being drawn at random from a standard deck of playing cards. If the chosen card is a 3, the customer’s order is free. (Note that the probability of drawing a 3 from a standard deck of playing cards is 4/52. ) Otherwise, the customer must pay the bill. Suppose that 250 customers place lunch orders on the first day of the promotion. Let X = the number of people who win a free lunch. Statistics and Probability with Applications, 3 rd Edition 10

LESSON APP 5. 4 1. 2. 3. 4. Free lunch? Explain why X is a binomial random variable. Find the mean of X. Interpret this value in context. Find the standard deviation of X. Interpret this value in context. One of the customers is surprised when only 10 people win a free lunch. Should the customer be surprised? Find the probability that 10 or fewer people win a free lunch by chance alone and use this result to support your answer. Statistics and Probability with Applications, 3 rd Edition 11

Analyzing Binomial Random Variables Learning Targets After this lesson, you should be able to: ü Calculate and interpret the mean and standard deviation of a binomial distribution. ü Find probabilities involving several values of a binomial random variable. ü Use technology to calculate cumulative binomial probabilities. Statistics and Probability with Applications, 3 rd Edition 12