CHAPTER 3 SPECIAL PROBABILITY DISTRIBUTIONS EXERCISES 1. An office has 10 dot matrix printers. Each requires a new ribbon approximately every seven weeks. If the stock clerk finds at the beginning of a certain week that there are only five ribbons in stock, what is the probability that the supply will be exhausted during that week? 2. In a 10-question true-false test: (a) What is the probability of getting all answers correct by guessing? (b) What is the probability of getting eight correct by guessing? 3. A basketball player shoots 10 shots and the probability of hitting is 0.5 on each shot. (a) What is the probability of hitting eight shots? (b) What is the probability of hitting eight shots if the probability on each shot is 0.6? (c) What are the expected value and variance of the number of shots hit if p = 0.5? 4. A four-engine plane can fly if at least two engines work. (a) If the engines operate independently and each malfunctions with probability q, what is the probability that the plane will fly safely? (b) A two-engined plane can fly if at least one engine works. If an engine malfunctions with probability q, what is the probability that the plane will fly safely? (c) Which plane is the safest? 5. (a) The Chevalier de Mere used to bet that he would get at least one 6 in four rolls of a die. Was this a good bet? (b) He also bet that he would get at least one pair of 6's in 24 rolls of two dice. What was his probability of winning this bet? (c) Compare the probability of at least one 6 when six dice are rolled with the probability of at least two 6's when 12 dice are rolled. 6. If the probability of picking a winning horse in a race is 0.2, and if X is the number of winning picks out of 20 races, what is: (a) P[X = 4]. (b) P[X ? 4]. (c) E(X) and Var(X). 7. If X ~ BIN(n, p), derive E(X) using Definition 2.2.3. 8. A jar contains 30 green jelly beans and 20 purple jelly beans. Suppose 10 jelly beans are selected at random from the jar. (a) Find the probability of obtaining exactly five purple jelly beans if they are selected with replacement. (b) Find the probability of obtaining exactly five purple beans if they are selected without replacement.
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Step 1: Calculate the probability of x being equal to 2 using the binomial formula: \[ P(X=2) = \binom{4}{2} \times (1-q)^2 \times q^{4-2} \] Show more…
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David N.
Suppose a jar contains ten balls: six red and four white. Two balls are then randomly selected without replacement from the jar, and the number of red balls is recorded as the outcome of a random variable X. Which of the following statements is correct? (a) X doesn't have a binomial distribution since n=6 and p=0.4. (b) X doesn't have a binomial distribution since n=10 and p=0.6. (c) There is an unequal number of red and white balls in the jar, and the balls are selected without replacement. For this reason, the probability of choosing a red ball changes from trial to trial. In a bag, there are 9 pieces of chocolate in total: 5 pieces of them are milk chocolate and 4 pieces are dark chocolate. Suppose that you have no preference for either and simply pick two pieces of chocolate out of the bag completely at random. Let X be the number of milk chocolate from your draw. (a) Compute P(X = 1). (b) Compute the mean of X. (c) Compute the variance of X. Sam is a very good darts player and can hit the bull's eye (red circle in the center of the dart board) 60% of the time. Calculate the following probabilities: (a) What is the probability that he hits the bullseye for the 10th time on the 15th try? (b) What is the probability that he hits the bullseye 10 times in 15 tries? (c) What is the probability that he hits the first bullseye on the third try? Houying and Thomas are going to play 10 games of Mario Kart against each other. Both Houying and Thomas are experienced players, so their chances of winning don't change from game to game. Houying is a better player, however, and has a 60% chance of winning any game against Thomas. You can assume all the games are independent. In the initial 6 games, Houying won 5 of them. Compute the probability that Houying will win 7 games in total out of 10, given Houying has already won 5 out of the initial 6 games. Compute the required probability.
Jon S.
1) The following is a valid probability distribution for a random variable X. What must the value for P(2) be? [0.25] X P(X) 0 0.15 1 0.20 2 ???? 3 0.40 2) Calculate the expected value for the given probability distribution. [8.7] X P(X) 6 0.15 8 0.35 10 0.50 3) A soft drink company has a promotion. Each cap liner for 1,000,000 bottles is printed with either a cash prize or a "Please Play Again". The distribution of each liner along with its frequency is shown below. What is the expected value of an individual liner? [6.3 cents] Liner # of Liners Play Again 949,779 $1.00 50,000 $5.00 200 $100.00 20 $10,000.00 1 4) A chocolate company makes candy-coated chocolate, 40% of which are red. The production line mixes the candy randomly and packages 10 per box. a) What is the probability that less than four candies in a given box are red? [0.3823] b) What is the probability that at least four candies in a given box are red? [0.6177] c) Describe a second way of finding the answer to part (b) 5) Prepare a table and graph for a binomial probability distribution with n = 5 and p = 0.5. 6) One type of jet engine has a 0.0001 probability of failure while in flight. For a jet that has four of these engines, what is the probability of at least two of them failing? [0.00000005999] 7) Suppose that 65% of the families in a town own computers. If ten families are surveyed at random, a) what is the probability that at least five own computers? [0.91] b) what is the expected number of families that own computers? [6.5] 8) Ninety percent of Canadians are right-handed. a) What is the probability that exactly 29 people in a group of 30 are right-handed? [0.14] b) What is the expected number of right-handed people in a group of 30? [27]
Adi S.
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