5. A survey of subscribers to Forbes showed that 72% have investments in money market funds (Forbes 1993 Subscriber Study). Further, of those who have invested in money market funds, 27.78% have also invested in certificates of deposit (CDs). Of those who have not invested in money market funds, 58.57% have invested in CDs. (a) Organize the above information into a probability tree. (b) What percentage of subscribers have invested in both money market funds and CDs? (c) What percentage of subscribers have invested in either money market funds or CDs?

6. A survey of benefits for 254 corporate executives (Business Week, October 24, 1994) showed that 155 executives were provided mobile phones, 152 were provided club memberships, and 110 were provided both mobile phones and club memberships as perks associated with their position. (a) Let M be the event of having a mobile phone and C be the event of having a club membership. Find the following probabilities: P(M ), P(C), and P(M C). (b) Use the probabilities in part (a) to compute the probability that a cor- porate executive has at least one of the two perks. (c) What is the probability that a corporate executive does not have either of these perks.

(d) Organize the probabilistic information into a probability tree whose first stage represents ownership of a mobile phone, and whose second stage represents enrollment in club membership. (e) Determine the probability that a corporate executive does not have club membership given that that executive has a mobile phone. (f ) Determine the probability that a corporate executive does not have a mobile phone given that that executive does not have club membership.

7. A consulting firm has submitted a bid for a large research project. The firm's management initially felt there was a 50-50 chance to getting the project. However, the agency to which the bid was submitted has subsequently re- quested additional information on the bid. Past experience indicates that on 75% of the successful bids and 40% of the unsuccessful bids the agency requested additional information. (a) Draw a probability tree where the first stage represents whether the bid ultimately succeed or fail and where the second stage represents whether or not additional information will be requested. (b) Flip the probability tree from part (a). (c) What is the prior probability of the bid being successful (that is, prior to the request for additional information)? (d) What is the conditional probability of a request for additional informa- tion given that the bid will be ultimately successful. (e) Use Bayes' Rule to compute a conditional probability that the bid will be successful given that a request for additional information has been received. (f) Are the events "successful bid" and "additional information requested" independent? Show how you know this.