1. Whistle Blowers. Although there is legal protection for â€œwhistle blowersâ€â€”employees who report illegal or unethical activities in the workplaceâ€”it has been reported that approximately 23% of those who reported fraud suffered reprisals such as demotion or poor performance ratings. Suppose the probability that an employee will fail to report a case of fraud is .69. Find the probability that an employee who observes a case of fraud will report it and will subsequently suffer some form of reprisal.

2. DVRs. A retailer sells two styles of digital video recorders (DVR) that are in equal demand. (Fifty percent of all potential customers prefer style 1, and 50% favor style 2.) If the retailer stocks four of each, what is the probability that the first four customers seeking a DVR all purchase the same style?

3. Interstate. Commerce A shipping container contains seven complex electronic systems. Unknown to the purchaser, three are defective. Two of the seven are selected for thorough testing and are then classified as defective or no defective. What is the probability that no defectives are found?

4. A Reticent Salesman. The probability that a salesperson makes a sale during the first contact with a client is .4 but improves to .55 on the second contact, given that the client did not buy during the first contact. This sales-person makes one and only one callback to a client. a. What is the probability that the client will buy? b. What is the probability that the client will not buy?

5. Rental trucks. A rental truck agency services its vehicles on a regular basis to check for mechanical problems. The agency has six moving vans, two of which have brake problems. During a routine check, the vans are tested one at a time.

a. What is the probability that the last van with brake problems is the fourth van tested?

b. What is the probability that no more than four vans need to be tested before both brake problems are detected?

c. Given that one van with bad brakes is detected in the first two tests, what is the probability that the remaining van is found on the third or fourth test?

6. Pennsylvania lottery. Probability played a role in the rigging of the April 24, 1980, Pennsylvania state lottery. To determine each digit of the three-digit winning number, each of the numbers 0, 1, 2, . . . , 9 was written on a Ping-Pong ball, the 10 balls were blown into a compartment, and the number selected for the digit was the one on the ball that floats to the top of the machine. To alter the odds, the conspirators injected a liquid into all balls used in the game except those numbered 4 and 6, making it almost certain that the lighter balls would be selected and determine the digits in the winning number. They then proceeded to buy lottery tickets bearing the potential winning numbers. How many potential winning numbers were there (666 was the eventual winner)?

7. Lottery, continued. Refer to Exercise 6. Hours after the rigging of the Pennsylvania state lottery was announced on September 19, 1980, Connecticut state lottery officials were stunned to learn that their winning number for the day was 666.

a. All evidence indicates that the Connecticut selection of 666 was pure chance. What is the probability that a 666 would be drawn in Connecticut, given that a 666 had been selected in the April 24, 1980, Pennsylvania lottery?

b. What is the probability of drawing a 666 in the April 24, 1980, Pennsylvania lottery (remember, this drawing was rigged) and a 666 on the September 19, 1980, Connecticut lottery?

8. The Birthday problem. Two people enter a room and their birthdays (ignoring years) are recorded.

a. Identify the nature of the simple events in S.

b. What is the probability that the two people have a specific pair of birthdates?

c. Identify the simple events in event A: Both people have the same birthday.

d. Find P(A)

e. Find P(A^c).

9. The Birthday problem, continued. If n people enter a room, find these probabilities:

A: None of the people have the same birthday B: At least two of the people have the same birthday

Solve for

A. n=3

B. n=4

[Note: Surprisingly, P(B) increases rapidly as n increases. For example, for n =20, P(B) =.411; for n=40, P(B)=.891.]

10. The American Journal of Sports Medicine published a study of 810 women collegiate rugby players with two common knee injuries: medial cruciate ligament (MCL) sprains and anterior cruciate ligament (ACL) tears^7. For backfield players, it was found that 39% had MCL sprains and 61% had ACL tears. For forwards, it was found that 33% had MCL sprains and 67% had ACL tears. Since a rugby team consists of eight forwards and seven backs, you can assume that 47% of the players with knee injuries are backs and 53% are forwards.

a. Find the unconditional probability that a rugby player selected at random from this group of players has experienced an MCL sprain.

b. Given that you have selected a player who has an MCL sprain, what is the probability that the player is a forward?

c. Given that you have selected a player who has an ACL tear, what is the probability that the player is a back?