Problem Set 07 Discrete Random Variables

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1 Name Problem Set 07 Discrete Random Variables MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the mean of the random variable. 1) The random variable X is the number of golf balls ordered by customers at a pro shop. Its probability distribution is given in the table. Round the answer to two decimal places when necessary. x P(X = x) A) 8.22 B) 5.55 C) 9 D) 9.3 1) 2) The random variable X is the number of siblings of a student selected at random from a particular secondary school. Its probability distribution is given in the table. Round the answer to three decimal places when necessary. x P(X = x) A) B) C) 2.5 D) ) Find the standard deviation of the random variable. 3) The random variable X is the number of people who have a college degree in a randomly selected group of four adults from a particular town. Its probability distribution is given in the table. Round the answer to two decimal places. x P(X = x) A) 0.84 B) 2.95 C) 0.92 D) ) 4) A police department reports that the probabilities that 0, 1, 2, and 3 burglaries will be reported in a given day are 0.48, 0.36, 0.14, and 0.02, respectively. Find the standard deviation for the probability distribution. Round the answer to two decimal places. A) 1.05 B) 0.78 C) 0.61 D) ) The probability distribution of a random variable is given along with its mean and standard deviation. Draw a probability histogram for the random variable; locate the mean and show one, two, and three standard deviation intervals. 1

2 5) x P(X = x) ) μ = 5.7, σ = 0.95 A) B) C) Find the expected value of the random variable. Round to the nearest cent unless stated otherwise. 6) In a game, you have a 1/33 probability of winning $62 and a 32/33 probability of losing $2. What is your expected value? A) -$1.94 B) $1.88 C) $3.82 D) -$0.06 6) 7) A contractor is considering a sale that promises a profit of $27,000 with a probability of 0.7 or a loss (due to bad weather, strikes, and such) of $2000 with a probability of 0.3. What is the expected profit? Round the answer to the nearest dollar. A) $18,900 B) $25,000 C) $18,300 D) $20,300 7) 2

3 8) Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $4.00 for rolling a 4 or a 5, nothing otherwise. What is your expected value? A) -$2.00 B) -$0.67 C) $4.00 D) $2.00 8) 9) Sue Anne owns a medium-sized business. Use the probability distribution below, where X describes the number of employees who call in sick on a given day. 9) Number of Employees Sick P(X = x) What is the expected value of the number of employees calling in sick on any given day? Round the answer to two decimal places. A) 1.00 B) 1.90 C) 2.00 D) 1.85 Evaluate the expression. 10) 9! 7! 10) A) 63,000 B) 2! C) 9 7 D) 72 11) 11 3 A) 165 B) 330 C) 40,320 D) 4 11) Determine the binomial probability formula given the number of trials and the success probability for Bernoulli trials. Let X denote the total number of successes. Round to three decimal places. 12) n = 5, p = 0.6, P(X = 2) 12) A) B) C) D) ) n = 4, p = 1, P(X = 3) 4 13) A) B) C) D) Find the specified probability distribution of the binomial random variable. 14) A multiple choice test consists of four questions. Each question has five possible answers of which only one is correct. A student guesses on every question. Find the probability distribution of X, the number of questions she answers correctly. A) B) C) D) ) 3

4 15) In one city, the probability that a person will pass his or her driving test on the first attempt is Four people are selected at random from among those taking their driving test for the first time. Determine the probability distribution of X, the number among the four who pass the test. A) B) C) D) ) 38% of the murder trials in one district result in a guilty verdict. Five murder trials are selected at random from the district. Determine the probability distribution of X, the number of trials among the five selected in which the defendant is found guilty. A) B) C) D) ) 16) Find the indicated binomial probability. Round to five decimal places when necessary. 17) In a certain college, 20% of the physics majors belong to ethnic minorities. If 10 students are selected at random from the physics majors, what is the probability that exactly 2 belong to an ethnic minority? A) 1.8 B) C) D) ) 18) A multiple choice test has 30 questions, and each has four possible answers, of which one is correct. If a student guesses on every question, find the probability of getting exactly 12 correct. A) B) C) 13,922,008.7 D) ) 19) A company manufactures calculators in batches of 64 and there is a 4% rate of defects. Find the probability of getting exactly 4 defects in a batch. A) 3.84 B) C) 54, D) ) 20) A cat has a litter of 7 kittens. Find the probability that exactly 5 of the little furballs are female. Assume that male and female births are equally likely. A) B) C) D) ) 4

5 Find the indicated probability. Round to four decimal places. 21) A machine has 7 identical components which function independently. The probability that a component will fail is 0.2. The machine will stop working if more than three components fail. Find the probability that the machine will be working. A) B) C) D) ) 22) Find the probability of at least 2 girls in 6 births. Assume that male and female births are equally likely and that the births are independent events. A) B) C) D) ) 23) An airline estimates that 93% of people booked on their flights actually show up. If the airline books 73 people on a flight for which the maximum number is 71, what is the probability that the number of people who show up will exceed the capacity of the plane? A) B) C) D) ) 24) A car insurance company has determined that 9% of all drivers were involved in a car accident last year. Among the 10 drivers living on one particular street, 3 were involved in a car accident last year. If 10 drivers are randomly selected, what is the probability of getting 3 or more who were involved in a car accident last year? A) B) C) D) ) 25) In one city, the probability that a person will pass his or her driving test on the first attempt is people are selected at random from among those taking their driving test for the first time. What is the probability that among these 11 people, the number passing the test is between 2 and 4 inclusive? A) B) C) D) ) 5

6 Construct a probability histogram for the binomial random variable, X. 26) Two balls are drawn at random, with replacement, from a bag containing 4 red balls and 2 blue balls. X is the number of blue balls drawn. A) B) 26) C) D) Find the mean of the binomial random variable. Round to two decimal places when necessary. 27) According to a college survey, 22% of all students work full time. Find the mean for the random variable X, the number of students who work full time in samples of size 16. A) 0.22 B) 3.52 C) 4 D) ) 28) A die is rolled 4 times and the number of times that two shows on the upper face is counted. If this experiment is repeated many times, find the mean for the random variable X, the number of twos. A) 0.67 B) 3.33 C) 1 D) ) 29) On a multiple choice test with 6 questions, each question has four possible answers, one of which is correct. For students who guess at all answers, find the mean for the random variable X, the number of correct answers. A) 2 B) 4.5 C) 1.5 D) 3 29) 30) The probability that a radish seed will germinate is 0.7. A gardener plants seeds in batches of 12. Find the mean for the random variable X, the number of seeds germinating in each batch. A) 3.6 B) 10.8 C) 8.4 D) ) 31) A company manufactures batteries in batches of 20 and there is a 3% rate of defects. Find the mean for the random variable X, the number of defects per batch. A) 19.4 B) 0.58 C) 0.6 D) ) 6

7 32) The probability that a person has immunity to a particular disease is 0.3. Find the mean for the random variable X, the number who have immunity in samples of size 24. A) 12 B) 0.3 C) 16.8 D) ) 33) The probability is 0.3 that a person shopping at a certain store will spend less than $20. For groups of size 16, find the mean number who spend less than $20. A) 14 B) 6 C) 4.8 D) ) 34) In a certain town, 90 percent of voters are in favor of a given ballot measure and 10 percent are opposed. For groups of 180 voters, find the mean for the random variable X, the number who oppose the measure. A) 10 B) 90 C) 18 D) ) Find the standard deviation of the binomial random variable. 35) According to a college survey, 22% of all students work full time. Find the standard deviation for the random variable X, the number of students who work full time in samples of size 16. A) 3.52 B) 1.88 C) 1.66 D) ) 36) A die is rolled 17 times and the number of twos that come up is tallied. If this experiment is repeated many times, find the standard deviation for the random variable X, the number of twos. A) 2.06 B) C) D) ) 37) On a multiple choice test with 10 questions, each question has four possible answers, one of which is correct. For students who guess at all answers, find the standard deviation for the random variable X, the number of correct answers. A) B) C) D) ) 38) The probability that a radish seed will germinate is 0.7. A gardener plants seeds in batches of 11. Find the standard deviation for the random variable X, the number of seeds germinating in each batch. A) B) C) D) ) 39) A company manufactures batteries in batches of 26 and there is a 3% rate of defects. Find the standard deviation for the random variable X, the number of defects per batch. A) B) C) 0.87 D) ) 40) The probability of winning a certain lottery is 1/70,366. For people who play 929 times, find the standard deviation for the random variable X, the number of wins. A) B) C) D) ) 7

8 Answer Key Testname: UNTITLED1 1) A 2) D 3) C 4) B 5) B 6) D 7) C 8) B 9) D 10) D 11) A 12) C 13) B 14) B 15) A 16) D 17) D 18) A 19) B 20) C 21) C 22) D 23) A 24) A 25) A 26) A 27) B 28) A 29) C 30) C 31) C 32) D 33) C 34) C 35) C 36) B 37) C 38) D 39) C 40) D 8

MATH 227 CP 6 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

MATH 227 CP 6 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Identify the given random variable as being discrete or continuous. 1) The number of phone