On a given weekend in the fall, a tire company can buy television and advertising time for a college football game, a baseball game, or a professional football game. If the company sponsors the college football game, there is a 70% chance of a high rating, a 40% chance if they sponsor a baseball game, and a 60% chance if they sponsor a professional football game. The probabilities of the company sponsoring these various games are 0.5, 0.2, and 0.3, respectively. Suppose the company does get a high rating, find the probability that it sponsored a professional football game. The probability is: (Type an integer or a simplified fraction.)

Three cowboys Doc Holiday, Wyatt Erp, and Wild Bill are locked in a 3-way duel. When the churchbell rings at high noon, all three will draw their guns and fire one shot simultaneously. A hit is registered if a shooter is hit by either of the other two cowboys.  Doc Holiday will randomly aim and fire at either Wyatt Erp or Wild Bill, with an even split probability of 0.5 of who he'll shoot at. Being a good shot, he has a 60% chance of hitting the target he fires at.  Wyatt Erp will randomly aim and fire at either Doc Holiday or Wild Bill, with an even split probability of 0.5 of who he'll shoot at. Also a good shot, he has a 60% chance of hitting the target he fires at.  Wild Bill will randomly aim and fire at either Wyatt Erp or Doc Holiday, with an even split probability of 0.5 of who he'll shoot at. Being a poor shot, he only has a 30% chance of hitting the target he fires at.  Let X denote the number of cowboys that are hit after all three take one shot at the same time. a) What are…

(QUEEN'S SECRET AND GENIES) In old days, there was a queen and she had a secret problem to solve. To get an answer to her secret problem, the queen bought one female and one male genie lamps. The probabilities that her secret problem can be solved by the male and female genies are 0.3 and 0.4 respectively. A Queen looking for divine help. Assume the queen keeps the genies independently away from each other with a fear that both could not harm her, what is the probability that the queen's secret problem is solved by at least one of the genies? Note: Please keep accuracy upto two decimal places.

Bob, the proprietor of Midway Lumber, bases his projections for the annual revenues of the company on the performance of the housing market. He rates the performance of the market as very strong, strong, normal, weak, or very weak. For the next year, Bob estimates that the probabilities for these outcomes are 0.15, 0.25, 0.39, 0.09, and 0.12, respectively. He also thinks that the revenues corresponding to these outcomes are $23, $18.6, $16.9, $14, and $6 million, respectively. What is Bob's expected revenue for next year?