HW 5 Binomial and Geometric Probability Distribution

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Feb 20, 2024

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Unit 4 – Probability, Random Variables, and Probability Distributions HW 5 – Binomial and Geometric Probability Distributions 1) Zach is a 65% free throw shooter and each shot he takes is independent. In his practice tonight, Zach is going to shoot 40 free throws. a) Verify that this situation is a binomial random variable by checking BINS. b) "Zach will make exactly 25 free throws"? What is this in probability notation? _____________ Find the probability. c) "Zach will make at most 30 free throws"? What is this in probability notation? _____________ Find the probability. d) What is the notation that represents "Zach will make at least 20 free throws"? What is this in probability notation? _____________ Find the probability. e) What is the notation that represents "Zach will less than 23 free throws"? What is this in probability notation? _____________ Find the probability.

2) Amarillo Slim, a professional dart player, has an 80% chance of hitting the bullseye on a dartboard with any throw. Suppose that he throws 10 darts, one at a time, at the dartboard. a) Find the probability that Slim hits the bullseye exactly six times. b) Find the probability that he hits the bullseye at least four times. (c) Compute the mean of the number of bullseyes in 10 throws. 3) A university claims that 80% of its basketball players get their degree. An investigation examines the fates of a random sample of 20 players who entered the program over a period of several years. Of these players, 10 graduated and 10 are no longer in school. If the university's claim is true, what is the probability that exactly 10 out of 20 graduate? Can you conclude anything about the university's claim? 4) A quarterback completes 55% of his passes. (a) What is the probability that the quarterback throws 3 incomplete passes before he has a completion? (b) What is the probability that the quarterback throws his first completion in no more than 3 attempts? (c) How many passes, on average, can the quarterback expect to throw before he completes his first pass?

5) Decide whether the situation described is a binomial setting, a geometric setting or neither. a) Suppose you repeatedly draw cards out of a deck without replacement in an attempt to draw an ace. b) You roll a die 30 times and count the number of 1’s you rolled. c) You attempt to find a parking space at the mall on Christmas eve, so you drive up the parking lot aisle until you find an empty parking space. d) In an advertisement, you see that 24% of Americans use Crest toothpaste. So you take a random sample of 50 Americans and count the number who uses Crest toothpaste. e) You flip a coin until you get a tails. f) 8% of the apples at the supermarket have a bruise on it. You randomly choose apples from the fruit bin at Woodman's until you find one with a bruise. g) You randomly choose 20 apples from the fruit bin and count the number of apples without a bruise. _____ 6) A carton of 12 eggs contains 4 that are spoiled. Three eggs are selected to make an omelet. Find the probability that 2 of the selected eggs are spoiled. (A) 0.218 (B) 0.027 (C) 0.333 (D) 0.188 _____ 7) What are the mean and standard deviation of a binomial experiment that occurs with probability of success 0.76 and is repeated 150 times? (A) 114, 27.35 (B) 100.5, 5.23 (C) 114, 5.23 (D) 100.5, 27.35 (E) The mean is 114, but there is not enough information given to determine the standard deviation.

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______ 8) Which of the following is not required in a binomial setting? (A) Each trial is considered either a success or failure. (B) Each trial is independent. (C) The value of the random variable of interest is the number of trials until the first success occurs. (D) There is a fixed number of trials. (E) Each trial succeeds or fails with the same probability. ______ 9) A dealer in the Sands Casino in Las Vegas selects 40 cards from a standard deck of 52 cards. Let Y be the number of red cards (hearts or diamonds) in the 40 cards selected. Which of the following best describes this setting: (A) Y has a binomial distribution with n = 40 observations and probability of success p = 0.5. (B) Y has a binomial distribution with n = 40 observations and probability of success p = 0.5, provided the deck is shuffled well. (C) Y has a binomial distribution with n = 40 observations and probability of success p = 0.5, provided after selecting a card it is replaced in the deck and the deck is shuffled well before the next card is selected. (D) Y has a normal distribution with mean p = 0.5. ______ 10) Twenty percent of all trucks undergoing a certain inspection will fail the inspection. Assume that trucks are independently undergoing this inspection, one at a time. The expected number of trucks inspected before a truck fails inspection is (A) 2 (B) 4 (C) 5 (D) 20 (E) The answer cannot be computed from the information given.

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