Exam 8: Introduction to Probability

A small town has 8500 adult males and 5500 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 drinks heavily DH? ​

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. ​ ​

If a pair of dice, one green and one red, is rolled, what is the probability that the sum of the two dice is a number between 5 and 9, inclusive? ​

Suppose that all license plates in a state have three of the letters A through Z followed by four of the digits 0 through 9. If a plate is chosen at random, what is the probability that all three letters and all four digits on the plate will be different? Round your answer to five decimal places. ​

A company estimates that 50% of the country has seen its commercial and that if a person sees its commercial, there is a 45% probability that the person will buy its product. What is the probability that a person chosen at random in the country will have seen the commercial and bought the product? Round your answer to four decimal places. ​

A red ball and 10 white balls are in a box. If two balls are drawn, without replacement, what is the probability of getting 2 white W balls? ​

If 11 people are qualified for the next flight to a space station, how many different groups of 3 people can be chosen for the flight? ​

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? ​

Find the steady-state vector associated with the transition matrix . Round all numerical values in your answer to two decimal places. ​

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. ​

Suppose a die is tossed 1,000 times and a 1 comes up 667 times. Find the empirical probability for a 1 to occur. On the basis of a comparison of the empirical probability and the theoretical probability, do you think the die is fair or biased? ​

A mathematics class consists of 9 engineering majors E, 14 science majors S, and 8 liberal arts majors LA. What is the probability that a student selected at random will be a science or liberal arts major? ​

The following table gives the percent of employees of the Ace Company in each of three salary brackets, categorized by the sex of the employees. An employee is selected at random. What is the probability that the person selected makes at least $50,000? ​ ​

If the probability of rain today is 23%, what are the odds against rain today? ​

A box of 11 transistors has 7 defective ones. If 2 transistors are drawn from the box together, what is the probability that both transistors are defective? Round your answer to two 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 bag contains 11 red balls and 13 white balls. Two balls are drawn without replacement. What is the probability that the second ball is white W, given that the first ball is red R? ​

A ball is drawn from a bag containing 14 red balls numbered 1-14 and 11 white balls numbered 15-25. What is the probability that the ball is red R or even-numbered E? ​