Exam 5: Discrete Probability Distributions

A tennis player makes a successful first serve 46% of the time. If she serves 8 times, what is the probability that she gets exactly 3 first serves in? Assume that each serve is independent of the others.

Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial. Round to three decimal places. - $\mathrm { n } = 4 , \mathrm { x } = 3 , \mathrm { p } = \frac { 1 } { 6 }$

Multiple-choice questions on a test each have 4 possible answers, one of which is correct. Assume that you guess the answers to 5 such questions. A) Use the multiplication rule to find the probability that the first 2 guesses are wrong and the last 3 guesses are correct. That is, find P(WWCCC), where C denotes a correct answer and W denotes a wrong answer. B) Make a complete list of the different possible arrangements of 2 wrong answers and 3 correct answers, then find the probability for each entry in the list. C) Based on the preceding results, what is the probability of getting exactly 3 correct answers when 5 guesses are made?

Suppose that computer literacy among people ages 40 and older is being studied and that the accompanying tables describes the probability distribution for four randomly selected people, where x is the number that are computer literate. Is it unusual to find four computer literates among four randomly selected people? () 0 0.16 1 0.25 2 0.36 3 0.15 4 0.08

Determine if the outcome is unusual. Consider as unusual any result that differs from the mean by more than 2 standard deviations. That is, unusual values are either less than $\mu - 2 \sigma \text { or greater than } \mu + 2 \sigma \text {. }$ -The Acme Candy Company claims that 60% of the jawbreakers it produces weigh more than .4 ounces. Suppose that 800 jawbreakers are selected at random from the production lines. Would it be unusual for this sample of 800 to contain 552 jawbreakers that weigh more than .4 ounces?

List the four requirements for a binomial distribution. Describe an experiment which is binomial and discuss how the experiment fits each of the four requirements.

Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial. Round to three decimal places. - $\mathrm { n } = 30 , \mathrm { x } = 12 , \mathrm { p } = 0.20$

Assume that a researcher randomly selects 14 newborn babies and counts the number of girls selected, x. The probabilities corresponding to the 14 possible values of x are summarized in the given table. Answer the question using the table. Probabilities of Girls x( girls) () ( girls) () ( girls) () 0 0.000 5 0.122 10 0.061 1 0.001 6 0.183 11 0.022 2 0.006 7 0.209 12 0.006 3 0.022 8 0.183 13 0.001 4 0.061 9 0.122 14 0.000 -Find the probability of selecting exactly 8 girls.

Assume that $\mathrm { x }$ is a random variable in a probability distribution with mean $\mu$ and standard deviation $\sigma$ . Find expressions for the mean and standard deviation if every value of $\mathrm { x }$ is modified by first being multiplied by 5 , then increased by 4 .

Find the mean of the given probability distribution. -The random variable x is the number of houses sold by a realtor in a single month at the Sendsom's Real Estate office. Its probability distribution is as follows. Houses Sold () Probability () 0 0.24 1 0.01 2 0.12 3 0.16 4 0.01 5 0.14 6 0.11 7 0.21

A gambler claimed that he had loaded a die so that it would hardly ever come up 1. He said the outcomes of 1, 2, 3, 4, 5, 6 would have probabilities $\frac { 1 } { 20 } , \frac { 1 } { 6 } , \frac { 1 } { 6 } , \frac { 1 } { 6 } , \frac { 1 } { 6 } , \frac { 1 } { 6 }$ respectively. Can he do what he claimed? Why or why not? Is a probability distribution described by listing the outcomes along with their corresponding probabilities? Why or why not?

Suppose that in one town 10% of people are left handed. Suppose that you want to find the probability of getting exactly 2 left-handed people when 4 people are randomly selected. Can the answer be found as follows: Use the multiplication rule to find the probability of getting two left handers followed by two right handers, which is (0.1)(0.1)(0.9)(0.9)? If not, explain why not and show how the required probability can be found.

Determine whether the given procedure results in a binomial distribution. If not, state the reason why. -Rolling a single "loaded" die 38 times, keeping track of the "fives" rolled.

Determine if the outcome is unusual. Consider as unusual any result that differs from the mean by more than 2 standard deviations. That is, unusual values are either less than µ - 2? or greater than µ + 2?. -According to AccuData Media Research, 36% of televisions within the Chicago city limits are tuned to "Eyewitness News" at 5:00 pm on Sunday nights. At 5:00 pm on a given Sunday, 2500 such televisions are randomly selected and checked to determine what is being watched. Would it be Unusual to find that 959 of the 2500 televisions are tuned to "Eyewitness News"?

Find the mean, µ, for the binomial distribution which has the stated values of n and p. Round answer to the nearest tenth. - $\mathrm { n } = 1589 ; \mathrm { p } = 0.57$

Determine whether the given procedure results in a binomial distribution. If not, state the reason why. -Spinning a roulette wheel 5 times, keeping track of the winning numbers.

Determine if the outcome is unusual. Consider as unusual any result that differs from the mean by more than 2 standard deviations. That is, unusual values are either less than $\mu - 2 \sigma \text { or greater than } \mu + 2 \sigma$ . -A survey for brand recognition is done and it is determined that 68% of consumers have heard of Dull Computer Company. A survey of 800 randomly selected consumers is to be conducted. For such groups of 800, would it be unusual to get 504 consumers who recognize the Dull Computer Company name?

A slot machine at a hotel is configured so that there is a 1/1200 probability of winning the jackpot on any individual trial. If a guest plays the slot machine 6 times, find the probability of exactly 2 jackpots. If a guest told the hotel manager that she had hit two jackpots in 6 plays of the slot machine, would the manager be surprised?

Find the indicated probability. -The brand name of a certain chain of coffee shops has a 57% recognition rate in the town of Coffleton. An executive from the company wants to verify the recognition rate as the company is interested in opening a coffee shop in the town. He selects a random sample of 10 Coffleton residents. Find the probability that exactly 4 of the 10 Coffleton residents recognize the brand name.

A contractor is considering a sale that promises a profit of $25,000 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?