Exam 8: Introduction to Probability

Suppose that five of the eight students running for class officers (president, vice president, secretary, and treasurer) have grade-point averages (GPAs) above 3.0. If the officers are selected at random, what is the probability that all four officers will be students with GPAs above 3.0? ​

A box of 27 transistors contains 7 that are defective. If 2 transistors are drawn from the box together, what is the probability that exactly one of the two transistors is defective? Round your answer to four decimal places.

In a certain city, the Democratic, Republican, and Consumer parties always nominate candidates for mayor. The probability of winning in any election depends on the party in power and is given by the following transition matrix. ​ ​ Using the given transition matrix and assuming the initial-probability vector is , find the probability vector for the fourth stage of the Markov chain. (This initial-probability vector indicates that a Consumer is certain to win the initial election.) ​

An automatic teller machine requires that each customer enter a five-digit personal identification number (PIN) when he or she inserts a bank card. If a person finds a bank card and guesses at a PIN to use the card fraudulently, and the person knows that the PIN will not have any digit repeated, what is the probability the person will succeed in guessing in one attempt? ​

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? ​

A bag contains 9 white balls and 5 red balls. Three balls are drawn, without replacement, from the bag. Construct a probability tree to determine the probability that all three balls are white W. ​

Suppose a fair coin is tossed 6 times. What is the probability of flipping exactly 2 heads? ​

A retailer purchases 140 of a new brand of DVD player, of which 4 are defective. The purchase agreement says that if he tests 8 chosen at random and finds 1 or more defective, he receives all the DVD players free of charge. What is the probability that he will not have to pay for the DVD players? Round your answer to three decimal places. ​

An urn contains 4 red, 7 white, and 9 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 both balls are white W. ​

A card is drawn from an ordinary deck of 52 playing cards. What is the probability that it will be a jack (J), queen (Q), or king (K)? ​

To see whether a bank has enough minority construction company loans, a social agency selects 31 loans to construction companies at random and finds that 4 of them are loans to minority companies. If the bank's claim that 10 of every 100 of its loans to construction companies are minority loans is true, what is the probability that 4 loans out of 31 are minority loans? Leave your answer with combination symbols. ​

A bag contains 7 white balls and 8 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn,with replacement, the first ball is red R and the second is white W. ​

Can the vector be a probability vector? ​

A ball is drawn from a bag containing 6 red balls numbered 1-6 and 14 white balls numbered 7-20. What is the probability that the ball is not red R? ​

One ball is drawn at random from a bag containing 17 red R balls and 5 white W balls. What is the probability that the ball is green G? ​

A mathematics class consists of 14 engineering majors E, 8 science majors S, and 15 liberal arts majors LA. What is the probability that a student selected at random will be an engineering or science 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 is male or makes less than $30,000? ​ ​

Suppose that an employer plans to hire four people from a group of eleven equally qualified people, of whom three are minority candidates. If the employer does not know which candidates are minority candidates, and if she selects her employees at random, what is the probability that no minority candidates are hired? Round your answer to two decimal places. ​

Suppose the following table summarizes the opinions of various groups on the issue of increased military spending. ​ ​ Given that a randomly selected individual is nonwhite NW, find the probability that he or she opposes increased military spending O. ​

If a state wants each of its license plates to contain 6 different digits, how many different license plates can it make if any of the digits 0 through 9 can appear in any of the 6 positions? ​