91Ó°ÊÓ

Consider the following addition rule to find the probability of the union of two events \(A\) and \(B\) : $$ P(A \text { or } B)=P(A)+P(B)-P(A \text { and } B) $$ When and why is the term \(P(A\) and \(B\) ) subtracted from the sum of \(P(A)\) and \(P(B) ?\) Give one example where you might use this formula.

The probability that a randomly selected elementary or secondary school teacher from a city is a female is \(.68\), holds a second job is \(.38\), and is a female and holds a second job is . 29 . Find the probability that an elementary or secondary school teacher selected at random from this city is a female or holds a second job.

The probability that a corporation makes charitable contributions is .72. Two corporations are selected at random, and it is noted whether or not they make charitable contributions. Find the probability that at most one corporation makes charitable contributions.

A company employs a total of 16 workers. The management has asked these employees to select 2 workers who will negotiate a new contract with management. The employees have decided to select these 2 workers randomly. How many total selections are possible? Considering that the order of selection is important, find the number of permutations.

A player plays a roulette game in a casino by betting on a single number each time. Because the wheel has 38 numbers, the probability that the player will win in a single play is \(1 / 38\). Note that each play of the game is independent of all previous plays. a. Find the probability that the player will win for the first time on the 10 th play. b. Find the probability that it takes the player more than 50 plays to win for the first time. c. A gambler claims that because he has 1 chance in 38 of winning each time he plays, he is certain to win at least once if he plays 38 times. Does this sound reasonable to you? Find the probability that he will win at least once in 38 plays.

Powerball is a game of chance that has generated intense interest because of its large jackpots. To play this game, a player selects five different numbers from 1 through 69 , and then picks a Powerball number from 1 through \(26 .\) The lottery organization randomly draws 5 different white balls from 69 balls numbered 1 through 69 , and then randomly picks a red Powerball number from 1 through \(26 .\) Note that it is possible for the Powerball number to be the same as one of the first five numbers. a. If a player's first five numbers match the numbers on the five white balls drawn by the lottery organization and the player's red Powerball number matches the Powerball number drawn by the lottery organization, the player wins the jackpot. Find the probability that a player who buys one ticket will win the jackpot. (Note that the order in which the five white balls are drawn is unimportant.)

A thief has stolen Roger's automatic teller machine (ATM) card. The card has a four-digit personal identification number (PIN). The thief knows that the first two digits are 3 and 5 , but he does not know the last two digits. Thus, the PIN could be any number from 3500 to \(3599 .\) To protect the customer, the automatic teller machine will not allow more than three unsuccessful attempts to enter the PIN. After the third wrong PIN, the machine keeps the card and allows no further attempts. a. What is the probability that the thief will find the correct PIN within three tries? (Assume that the thief will not try the same wrong PIN twice.) b. If the thief knew that the first two digits were 3 and 5 and that the third digit was either 1 or 7 , what is the probability of the thief guessing the correct PIN in three attempts?

A screening test for a certain disease is prone to giving false positives or false negatives. If a patient being tested has the disease, the probability that the test indicates a (false) negative is .13. If the patient does not have the disease, the probability that the test indicates a (false) positive is 10 . Assume that \(3 \%\) of the patients being tested actually have the disease. Suppose that one patient is chosen at random and tested. Find the probability that a. this patient has the disease and tests positive b. this patient does not have the disease and tests positive c. this patient tests positive d. this patient has the disease given that he or she tests positive