Exam 6: Discrete Probability Distributions

Interpret the Mean of a Discrete Random Variable as an Expected Value -A dice game involves throwing three dice and betting on one of the six numbers that are on the dice. The game costs $\$ 7$ to play, and you win if the number you bet appears on any of the dice. The distribution for the outcomes of the game (including the profit) is shown below: Number of dice with your number Profit Probability of Observing 0 -\ 7 125/216 1 \ 7 75/216 2 \ 9 15/216 3 \ 21 1/216 Find your expected profit from playing this game.

Compute the Mean and Standard Deviation of a Binomial Random Variable -A recent survey found that 70% of all adults over 50 wear sunglasses for driving. In a random sample of 60 adults over 50, what is the mean and standard deviation of those that wear sunglasses

Compute the Probabilities of Hypergeometric Experiments -A university must choose a team of 6 students to participate in a TV quiz show. The students will be chosen from a pool of 32 potential participants of whom 14 are women. If the 6 students are chosen at random, what is the probability that the team will contain 2 women

Construct Binomial Probability Histograms -Given that a random variable x, the number of successes, follows a Poisson process, the probability of 2 or more successes in any sufficiently small subinterval is

Compute the Mean and Standard Deviation of a Hypergeometric Random Variable -A manufacturer receives a shipment of 170 laptop computers of which 9 are defective. To test the shipment, the quality control engineer randomly selects 10 computers from the shipment and tests them. The random variable X represents the number of defective computers in the sample. What are the mean and standard deviation of the random variable X

Compute Probabilities of Binomial Experiments -A recent survey found that 70% of all adults over 50 wear sunglasses for driving. In a random sample of 10 adults over 50, what is the probability that at least six wear sunglasses

Compute Probabilities of a Poisson Random Variable -The number of goals scored at State College hockey games follows a Poisson distribution with a mean of 7 goals per game. Find the probability that a randomly selected State College hockey game would have more than three goals scored.

Compute Probabilities of Binomial Experiments -Sixty-five percent of men consider themselves knowledgeable soccer fans. If 14 men are randomly selected, find the probability that exactly four of them will consider themselves knowledgeable fans.

Compute Probabilities of Binomial Experiments -Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of exactly nine girls in ten births.

Compute the Variance and Standard Deviation of a Discrete Random Variable -The owner of 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 oranges. Determine the variance of the 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

Compute Probabilities of a Poisson Random Variable -The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.2. Find the probability of observing exactly three accidents on this stretch of road next month.

Compute the Probabilities of Hypergeometric Experiments -An electronics store receives a shipment of 54 flat screen TVs of which 11 are defective. During the quality control inspection, 3 TVs are selected at random from the shipment for testing. The shipment will only be accepted if all 3 TVs pass the inspection. What is the probability that the shipment will be accepted

Interpret the Mean of a Discrete Random Variable as an Expected Value -In a carnival game, a person wagers $2 on the roll of two dice. If the total of the two dice is 2, 3, 4, 5, or 6 then the person gets $4 (the $2 wager and $2 winnings). If the total of the two dice is 8, 9, 10, 11, or 12 then the person gets nothing (loses $2). If the total of the two dice is 7, the person gets $0.75 back (loses $0.25). What is the expected value of playing the game once

Classify the following random variable according to whether it is discrete or continuous -the number of emails received on any given day

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 standard deviation of the number of tickets written per day by the campus police

Compute Probabilities of a Poisson Random Variable -A dictionary contains 500 pages. If there are 200 typing errors randomly distributed throughout the book, use the Poisson distribution to determine the probability that a page contains exactly three errors.

Identify Discrete Probability Distributions -Consider the discrete probability distribution to the right when answering the following question. Find the probability that $x$ exceeds 5 . 3 5 7 9 () 0.26 ? 0.21 0.17

Interpret the Mean of a Discrete Random Variable as an Expected Value -The expected value of a discrete probability distribution may be negative.

Compute the Mean and Standard Deviation of a Binomial Random Variable -Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Suppose that 400 couples each have a baby; find the mean and standard deviation for the number of boys in the 400 babies.

Compute Probabilities of a Poisson Random Variable -A customer service firm receives an average of three calls per hour on its toll-free number. For any given hour, find the probability that it will receive at least three calls. Use the Poisson distribution.