PRACTICE EXERCISES - WEEK 5 1. The following table is a partial probability distribution for a company's projected profits (x = profit in $1000s) for the first year of operation (the negative value denotes a loss). x P(X = x) -100 0.10 0 0.20 50 0.30 100 0.25 150 0.10 200 ??? (a) What is P(X = 200)? (b) What is the probability that the company will be profitable? (c) Compute X =E(X) and interpret your answer. (d) Compute X . 2. Which of the following represent probability distributions for some random variable X and which do not? Give the reason(s) for your answer in each case. (a) x P(X=x) -2 1/10 0 3/10 4 5/10 7 1/10 (b) x P(X=x) 2 -1/4 3 1/4 5 3/4 7 1/4 (c) P(X = x) = x/6 for x = 0, 1, 2, 3 (d) P(X = x) = x/3 for x = 1, 2, 3 3. Suppose that the probability distribution of a random variable X can be described by = = the formula P ( X x ) (2 x 3) 2 for x = 0, 1, 2, 3, 4. 45 (a) Write out the probability distribution of X as a probability table. (b) Compute X (c) Compute X . 4. The probability that Abbas successfully completes a free throw in a basketball game is 0.7. Suppose that Abbas attempts 6 free throws in one game and that the outcomes of the free throws are independent. What is the probability that Abbas will successfully complete at least 5 free throws? 5. The worldwide number of airplane crashes follows a Poisson distribution with an average of three crashes per month. What is the probability that there will be (a) at least two such accidents in the next month (b) exactly five accidents in the next two months? 6. In a certain region, there are 25 animals, 5 of them are tagged. Suppose that 10 animals from that region are selected at random. Find the probability that (a) none of them are tagged. (b) at least two of them are tagged. 7. You randomly select 5 cards from a deck of 52 cards. Let the random variable X represent the total number of diamonds that you obtain. (a) If you sample the cards without replacement, what is the probability distribution of X? What is the probability that at most 2 of the cards will be diamonds? (b) If you sample the cards with replacement, what is the probability distribution of X? What is the probability that at most 2 of the cards will be diamonds? 8. Two balls are randomly chosen without replacement from an urn containing 4 white and 2 black balls. Suppose that we win $2 for each black ball selected, and we lose $1 for each for each white ball selected. Let X denote our winnings. (a) Write down the probability distribution of X as a probability table. (b) What is the expected winning amount? 9. The number of flaws on a VHS magnetic tape produced continuously at a factory follows a Poisson distribution with an average of 0.01 flaws per meter. A standard VHS cassette tape contains 250 meters of magnetic tape. What is the probability that there are at least two flaws in a single VHS cassette tape? 10. In a workforce of 10,000,000 people, there are 2,500,000 who are currently unemployed. If we take a sample of 10 people, what is the approximate probability that exactly four of them will be unemployed? 11. A graduate statistics course has six male and four female students. The professor wants to select two students at random to help her conduct a research project. Let X represent the number of female students chosen. (a) What is the probability distribution of X? (b) What is the probability that the two students chosen are female? 12. A hotel finds that 8% of all people that make a room reservation will cancel the reservation. If the hotel has 50 rooms available for a specific day and makes 53 reservations for that day, what is the probability that there will be a room available for every person who keeps their reservation? You may assume that cancellations occur independently of each other. SOLUTIONS 0.70 1. (a) 0.05 (b) P ( X 50) = (c) X = 55; The long-run average profit for the first year of operation for all such companies is $55,000. (d) X 73.993 2. (a) Valid. (b) Not valid. P(X = 2) = -1/4 < 0. (c) Valid. (d) Not valid. The sum of all probabilities must equal 1. 3. (a) x P(X = x) 0 9/45 1 1/45 2 1/45 3 9/45 4 25/45 2 (b) X 2.8889 (c) X 1.8543 4. 0.4202 5. (a) 0.8009 (b) 0.1606 6. Let X represent the number of tagged animals. X ~ H(N = 25, K = 5, n = 10). (a) P(X = 0) = 0.0565 (b) P(X 2) = 0.6866 7. (a) X ~H(N = 52, K = 13, n = 5), P(X 2) 0.907 (b) X ~ B(n = 5, p = 13/52 = 0.25), P(X 2) 0.897 8. (a) x P(X = x) -2 6/15 1 8/15 4 1/15 (b) 0 9. Let X represent the number of flaws in a randomly chosen VHS cassette tape. = = Thus, X ~ Poisson( 0.01(250) 2.5 ) and P(X 2) = 0.7127 10. 0.1460 11. (a) X~H(N = 10, K = 4, n = 2) (b) 0.1333 12. Let X represent the number of people who cancel their reservation. X~B(53,0.08). P(X 3) = 0.8070