Assume you have had your name submitted for both a team performance bonus (T) and a safety bonus (S). You know the chances of receiving both bonuses are pretty slim based on last year's results. All bonus money comes out of the corporate fund and the budgeted amount for this fund cannot be exceeded. There is also a limit on the total bonus money any one employee can receive. The criteria for qualifying for each bonus are different. The probability that you receive a safety bonus is 0.13. The probability of receiving both bonuses is 0.09. The probability of getting at least one of the bonuses is 0.3.
a) What is the "plain old vanilla" (unconditional) probability that you will receive a team bonus?
b) You realize that based on historical data the probability of receiving the team bonus seems higher if you have already received the safety bonus. Based on the information you have, calculate the actual conditional probability of receiving the team bonus given that you have already been awarded the safety bonus. What do you think about these results?
c) What is the probability that you are not offered any bonuses? Use a Venn diagram to explain.