Exam 6: Discrete Probability Distributions

Compute Probabilities of Binomial Experiments -A history professor decides to give a 10-question true-false quiz. She wants to choose the passing grade such that the probability of passing a student who guesses on every question is less than 0.10. What score should be set as the lowest passing grade

Interpret the Mean of a Discrete Random Variable as an Expected Value -In the American version of the Game Roulette, a wheel has 18 black slots, 8 red slots and 2 green slots. All slots are the same size. In a carnival game, a person wagers $2 on the roll of two dice. A person can wager on either red or black. Green is reserved for the house. If a player wagers $5 on either red or black and that color comes up, they win $10 otherwise they lose their wager. What is the expected value of playing the game once

Compute the Probabilities of Hypergeometric Experiments -In a lottery, a player selects six numbers between 1 and 39 inclusive. The six winning numbers (all different) are selected at random from the numbers 1-39. To win a prize, the player must match three or more of the winning numbers. What is the probability that the player matches exactly 3 numbers

Compute Probabilities of Binomial Experiments -A recent article in the paper claims that government ethics are at an all-time low. Reporting on a recent sample, the paper claims that 30% of all constituents believe their representative possesses low ethical standards. Suppose 20 of a representative s constituents are randomly and independently sampled. Assuming the paper s claim is correct, find the probability that more than eight but fewer than 12 of the 20 constituents sampled believe their representative possesses low ethical standards.

Compute the Mean and Standard Deviation of a Binomial Random Variable -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 40 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), how many of the 40 cards do we expect the psychic to guess correctly

Construct Probability Histograms -Calculate the mean for the discrete probability distribution shown here. 2 7 11 12 () .2 .3 .3 .2

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 6.5 tickets per day. Interpret the value of the mean.

Construct Binomial Probability Histograms -Given that a random variable x, the number of successes, follows a Poisson process, then the number of successes in any interval is independent of the number of successes in any other interval provided the intervals

Construct Probability Histograms -The produce manager at a farmer's market was interested in determining how many oranges a person buys when they buy oranges. He asked the cashiers over a weekend to count how many oranges a person bought when they bought oranges and record this number for analysis at a later time. The data is given below in the table. The random variable $x$ represents the number of oranges purchased and $P ( x )$ represents the probability that a customer will buy $x$ apples. Determine the mean number of oranges purchased by a customer. x 1 2 3 4 5 6 7 8 9 10 P(x) 0.05 0.19 0.20 0.25 0.12 0.10 0 0.08 0 0.01