Exam 6: Discrete Probability Distributions

A random variable is

Compute Probabilities of Binomial Experiments -A motel has a policy of booking as many as 150 guests in a building that holds 140. Past studies indicate that only 85% of booked guests show up for their room. Find the probability that if the motel books 150 guests, not enough seats will be available.

Compute the Probabilities of Hypergeometric Experiments -A IRS auditor randomly selects 3 tax returns from 50 returns of which 14 contain errors. What is the probability that none of the returns she selects contains an error

Compute the Variance and Standard Deviation of a Discrete Random Variable -A seed company has a test plot in which it is testing the germination of a hybrid seed. They plant 50 rows of 40 seeds per row. After a two-week period, the researchers count how many seed per row have sprouted. They noted that least number of seeds to germinate was 33 and some rows had all 40 germinate. The germination data is given below in the table. The random variable x represents the number of seed in a row that germinated and P(x) represents the probability of selecting a row with that number of seed germinating. Determine the standard deviation of the number of seeds per row that germinated. x 33 34 35 36 37 38 39 40 P(x) 0.02 0.06 0.10 0.20 0.24 0.26 0.10 0.02

Compute the Probabilities of Hypergeometric Experiments -A jury is to be selected from a pool of 34 potential jurors. The defendant faces the death penalty if convicted. Of the potential jurors, 8 are opposed to the death penalty and would not convict regardless of the evidence. The prosecutor knows that if even one juror opposes the death penalty, they will have no chance of getting a conviction. If none of the jurors opposes the death penalty they will have a chance of getting a conviction. What is the probability that none of the jurors opposes the death penalty, if the jury consists of 12 randomly selected jurors

Compute Probabilities of a Poisson Random Variable -The university police department must write, on average, five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 5.2 tickets per day. Find the probability that exactly nine tickets are written on a randomly selected day from this distribution.

Compute Probabilities of Binomial Experiments -In a recent survey, 70% of the community favored building a health center in their neighborhood. If 14 citizens are chosen, find the probability that exactly 6 of them favor the building of the health center.

Determine Whether a Probability Experiment is a Hypergeometric Experiment -A university must choose a team of 5 students to participate in a TV quiz show. The students will be chosen at random from a pool of 36 potential participants of whom 29 are women. The random variable $\mathrm { X }$ represents the number of women on the team.

Compute the Mean and Standard Deviation of a Binomial Random Variable -The probability that a football game will go into overtime is 14%. In 140 randomly selected football games, what is the mean and the standard deviation of the number that went into overtime

Determine Whether a Probability Experiment is a Binomial Experiment -Decide whether the experiment is a binomial experiment. If it is not, explain why. Testing a cough suppressant using 160 people to determine if it is effective. The random variable represents the number of people who find the cough suppressant to be effective.

Construct Binomial Probability Histograms -Draw the probability histogram and label the mean for n = 7 and p = 0.5

Compute Probabilities of a Poisson Random Variable -The university police department keeps track of the number of tickets it write in a year. Last year the campus police wrote 1460 tickets. Ticket writing on campus follows a Poisson process. What is the mean number of tickets written per day by the campus police

Construct Binomial Probability Histograms -Draw the probability histogram and label the mean for n = 6 and p = 0.4

Compute Probabilities of Binomial Experiments -A psychic network received telephone calls last year from over 1.5 million people. A recent article attempts to shed some light onto the credibility of the psychic network. One of the psychic network s psychics agreed to take part in the following experiment. Five different cards are shuffled, and one is chosen at random. The psychic will then try to identify which card was drawn without seeing it. Assume that the experiment was repeated 15 times and that the results of any two experiments are independent of one another. If we assume that the psychic is a fake (i.e., they are merely guessing at the cards and have no psychic powers), find the probability that they guess at least three correctly.

Determine Whether a Probability Experiment is a Binomial Experiment -Decide whether the experiment is a binomial experiment. If it is not, explain why. In a game you spin a wheel that has 14 different letters 850 times. The random variable represents the selected letter on each spin of the wheel.

Construct Probability Histograms -A seed company has a test plot in which it is testing the germination of a hybrid seed. They plant 50 rows of 40 seeds per row. After a two-week period, the researchers count how many seed per row have sprouted. They noted that least number of seeds to germinate was 33 and some rows had all 40 germinate. The germination data is given below in the table. The random variable $x$ represents the number of seed in a row that germinated and $\mathrm { P } ( \mathrm { x } )$ represents the probability of selecting a row with that number of seed germinating. Determine the mean number of seeds per row that germinated. x 33 34 35 36 37 38 39 40 P(x) 0.02 0.06 0.10 0.20 0.24 0.26 0.10 0.02

Compute the Mean and Standard Deviation of a Binomial Random Variable -In a recent survey, 80% of the community favored building a health center in their neighborhood. If 15 citizens are chosen, what is the standard deviation of the number favoring the health center

Interpret the Mean of a Discrete Random Variable as an Expected Value -Mamma Temte bakes six pies a day that cost $\$ 2$ each to produce. On $31 \%$ of the days she sells only two pies. On $39 \%$ of the days, she sells 4 pies, and on the remaining $30 \%$ of the days, she sells all six pies. If Mama Temte sells her pies for $\$ 5$ each, what is her expected profit for a day's worth of pies? [Assume that any leftover pies are given away.]

Interpret the Mean of a Discrete Random Variable as an Expected Value -A local bakery has determined a probability distribution for the number of cheesecakes that they sell in a given day. The distribution is as follows: Number sold in a day 0 5 10 15 20 Prob (Number sold) 0.1 0.15 0.19 0.16 0.4 Find the number of cheesecakes that this local bakery expects to sell in a day.

Compute Probabilities of a Poisson Random Variable -A small life insurance company has determined that on the average it receives 4 death claims per day. Find the probability that the company receives at least seven death claims on a randomly selected day.