Exam 7: Introduction to Probability

An urn contains 7 red, 3 white, and 6 black balls. One ball is drawn from the urn, it is replaced, and a second ball is drawn. Construct a probability tree to determine the probability that one ball is white W and one is red R. ​

A box contains 7 red balls, 6 white balls, and 3 black balls. Two balls are drawn at random from the box without replacement of the first before the second is drawn. What is the probability of getting a red R ball on the first draw and a white W ball on the second? ​

Suppose that a marksman hits the bull's-eye 1 time in 40 shots. If the next 4 shots are independent, find the probability that all of the next 4 shots hit the bull's-eye. ​

A frustrated store manager is asked to make seven different yes-no decisions that have no relation to each other. Because he is impatient to leave work, he flips a coin for each decision. If the correct decision in each case was no, what is the probability that all of his decisions were correct? ​

An ordinary die is tossed. What is the probability of getting a 6 or a 5? ​

The 2003 Toyota Matrix was first made available with 2 different engines and with 4 levels of trim, plus either front-wheel or all-wheel drive. If we include color and there were 8 colors available, how many different Toyota Matrix cars were available for purchase? ​

A local business A has two competitors, B and C. No customer patronizes more than one of these businesses at the same time. Initially, the probabilities that a customer patronizes A, B, or C are 0.4, 0.4, and 0.2, respectively. Suppose A initiates an advertising campaign to improve its business and finds the following transition matrix to describe the effect. ​ ​ ​ If A runs the advertising campaign for 3 weeks, find the probability that a customer will still patronize business B. ​

A company selling substandard drugs to developing countries sold 2,500,000 capsules with 150,000 of them empty. What is the probability that a person who takes a randomly chosen capsule from this company will get an empty capsule? ​

A sailboat owner received 9 different signal flags with his new sailboat. If the order in which the flags are arranged on the mast determines the signal being sent, how many 4-flag signals can be sent? ​

An urn contains five red R balls numbered 1, 2, 3, 4 and 5, five white W balls numbered 6, 7, 8, 9 and 10, and three black B balls numbered 11, 12 and 13. A ball is drawn from the urn. What is the probability that it is not black? ​

A bag contains 13 red balls and 11 white balls. Two balls are drawn without replacement. What is the probability that the second ball is red R, given that the first ball is white W? ​

Can the vector be a probability vector? ​

As a reward for a record year, the Ace Software Company is randomly selecting 5 people from its 70 employees for a free trip to Hawaii, but it will not pay for a traveling companion. If John and Jill are married and both are employees, what is the probability that they will both win? Round your answer to five decimal places. ​

Fifty-seven percent of marijuana use among youth occurs in the inner cities. If an instance of such marijuana use is chosen at random, what is the probability, rounded to two decimal places, that the use occurs in an inner city? ​

A small town has 6500 adult males and 5000 adult females. A sociologist conducted a survey and found that 40% of the males and 30% of the females drink heavily. An adult is selected at random from the town. What is the probability, rounded to four decimal places, that the person is male M? ​

Suppose a government study estimated that the probability of successive generations of a rural family remaining in a rural area was 0.4 and the probability of successive generations of an urban family remaining in an urban area was 0.6. Assuming a Markov chain applies to these facts, find the steady-state vector. ​

The following table gives the numbers of AIDS deaths in a single year for people over age 13 in various categories. Use the table to find the probability, rounded to four decimal places, that a person who died of AIDS in this year was male or black (non-Hispanic). ​

For species that reproduce sexually, characteristics are determined by a gene from each parent. Suppose that for a certain trait there are two possible genes available from each parent: a dominant gene D and a recessive gene r. Then the different gene combinations (called genotypes) for the offspring are DD, Dr, rD, and rr, where Dr and rD produce the same trait. Suppose further that these genotypes are the states of a Markov chain with the transition matrix given below. Find the steady-state vector for this genotype. Round all numerical values in your answer to two decimal places. ​ ​

A man owns an Audi, a Ford, and a VW. He drives everyday and never drives the same car two days in a row. These are the probabilities that he drives each of the other cars the next day: ​ Pr(Ford after Audi) = 0.5 Pr(VW after Audi) = 0.5 Pr(Audi after Ford) = 0.6 Pr(VW after Ford) = 0.4 Pr(Audi after VW) = 0.9 Pr(Ford after VW) = 0.1 ​ If he drove the Ford on Monday, what is the probability that he will drive the VW on Wednesday? ​

Can the vector be a probability vector?