Statistics and Probability Answers

9. Terry is a typical investment advisor at BeatTheMarket.com. Each stock that Terry recommends outperforms the stock market average over the upcoming year with probability 0.5, independent of all other stocks and all other recommendations he or any other advisor makes.

George is a good investment advisor at BeatTheMarket.com. Each stock that George recommends outperforms the stock market average over the upcoming year with probability 0.6, independent of all other stocks and all other recommendations he or any other advisor makes. BeatTheMarket.com gives you a list of 100 stocks that are all recommended by one of their two advisors. The company tells you that the list is equally likely to have come from Terry or from George.

(b) Given that 60 or more of the stocks on the list outperform the market over the upcoming year, what’s the probability that the list came from George.

(Hint: use Bayes’ rule)

9. Terry is a typical investment advisor at BeatTheMarket.com. Each stock that Terry recommends outperforms the stock market average over the upcoming year with probability 0.5, independent of all other stocks and all other recommendations he or any other advisor makes.

(a) You ask Terry to pick 100 different stocks. What’s the probability that 60 or more of the stocks he recommends will outperform the stock market average over the next year. (Use normal approximation)

7. Parking your car at MIT is expensive, so expensive that you decide that it must be cheaper to keep your car parked illegally in a tow zone and chance the occasional ticket or tow. On each day, a Cambridge meter maid will notice your illegally parked car with probability 0.25. Upon noticing your illegally parked car, with probability 0.8 he or she will only issue you a ticket; otherwise your car will be towed. All of this occurs independently on each day, and independent of what happens on other days.

Suppose that in addition to the city of Cambridge, your car may also be towed by MIT. On each day with probability 0.1, an MIT tow truck will pass by your ‘parking space,’ independently of all other days. If you car has not yet been towed by the city of Cambridge, the MIT tow truck will tow away your car. Assume that MIT operates independently of Cambridge’s meter maids.

(d) What is the probability distribution of the time (in days) till your car is towed a seventh time?

7. Parking your car at MIT is expensive, so expensive that you decide that it must be cheaper to keep your car parked illegally in a tow zone and chance the occasional ticket or tow. On each day, a Cambridge meter maid will notice your illegally parked car with probability 0.25. Upon noticing your illegally parked car, with probability 0.8 he or she will only issue you a ticket; otherwise your car will be towed. All of this occurs independently on each day, and independent of what happens on other days.

(a) What is the expected time (in days) between successive times your car is towed?

(b) What is the standard deviation of the time (in days) between successive times your car is towed?

(c) What is the expected number of times your car is towed over the course of two months (60 days)?

6. Tina Fey goes trick-or-treating the night of Halloween dressed as Sarah Palin. She visits homes in her neighborhood to collect candy, but only receives candy, naturally, when the door is answered and the family still has a piece of candy to give away. Upon knocking, the probability of the door being answered is 3/4, and the probability that the home still has candy is 2/3. Assume that the events “Door answered” and “candy remaining” are independent and also that the outcomes at each home are independent. Also assume that each home gives away at most a single piece of candy.

(c) Determine the probability that she receives her second piece of candy at the fifth house.

(d) Given that she did not receive her second piece of candy at the second house, determine the conditional probability that she will receive her second piece of candy at the fifth house.

6. Tina Fey goes trick-or-treating the night of Halloween dressed as Sarah Palin. She visits homes in her neighborhood to collect candy, but only receives candy, naturally, when the door is answered and the family still has a piece of candy to give away. Upon knocking, the probability of the door being answered is 3/4, and the probability that the home still has candy is 2/3. Assume that the events “Door answered” and “candy remaining” are independent and also that the outcomes at each home are independent. Also assume that each home gives away at most a single piece of candy.

(a) Determine the probability that Tina receives her first piece of candy at the third house she visits.

(b) Given that she has received exactly four pieces of candy from the first eight houses, determine the conditional probability that Tina will receive her fifth piece of candy at the eleventh house.

3. A production process produces electronic component parts. It is presumed that the probability of a defective part is 0.01. During a test of this presumption, 600 parts are sampled randomly and 18 defectives are observed.

(a) What is your response to the presumption that the process is 1% defective? Be sure that a computed probability accompanies your comment.

(b) Under the presumption of a 1% defective process, what is the probability that only 3 parts will be found defective?

(c) Do parts (a) and (b) again using the Poisson approximation.