Elementary Statistics—More Probability and Random Variable Problems

1. Of the student body at a certain college, 52% are women. Furthermore, 5% of the students are majoring in computer science. Finally, 2% of the students are women majoring in computer science. Suppose a student is randomly chosen.
1. Given that the student a women, what is the conditional probability that she’s majoring in computer science?

2. Given that the student is a computer science major, what is the conditional probability that the student is a woman?

3. Are the events {woman} and {computer science major} independent? Why or why not?

2. Two shipping services offer overnight delivery of packages and both promise delivery before 9 a.m. A mail-order catalog company ships 75% of its overnight packages using shipping service 1 and 25% using shipping service 2. When service 1 is used, 2% of the time the 9 a.m. delivery promise is not met. Whereas, when service 2 is used, 5% of the time the 9 a.m. delivery promise is not met. Given that a package is late (i.e., arrives after 9 a.m.), what is the conditional probability it came through shipping service 2?

3. Bubba, always in search of new money sources, creates a game. In Bubba’s game, a player rolls a fair 4-sided die at the same time that Bubba rolls a fair 4-sided die. If the player rolls a higher number than Bubba, then Bubba pays the player $8 (note: the faces of each die are labeled “1”, “2”, “3”, “4”). Otherwise, Bubba pays the player nothing. How much should Bubba charge for each play of the game so his expected net profit is $2?

4. A local restaurant accepts either the American Express or the VISA credit card. A total of 24% of its customers carry an American Express card, 61% carry a VISA card, and 11% carry both. What percent of its customers carry at least one credit card that the restaurant accepts?

5. A simplified model for the movement of the price of a certain stock supposes that on each day the stock’s price either moves up 1 unit with probability 0.6 or it moves down 1 unit with probability 0.4. The changes on different days are assumed to be independent.
1. What is the probability that after 2 days the stock is at its original price?

2. What is the probability that after 3 days the stock’s price will have increased by 1 unit?

6. A health study tracked a group of people for five years. At the beginning of the study, 35% were classified as smokers and 65% were classified as nonsmokers. Results of the study showed that smokers were twice as likely to die as nonsmokers during the five-year study. Given that a randomly selected participant dies over the five-year period, determine the conditional probability the participant was a smoker.

7. Mark and Raul compete in an obstacle course race. Let X be Mark’s time to complete the course. From past experience, the mean and standard deviation of Mark’s times are 68.4 seconds and 4.2 seconds, respectively. Let Y be Raul’s time to complete the course. From past experience, the mean and standard deviation of Raul’s times are 64.9 seconds and 3.7 seconds, respectively. Mark and Raul compete individually, but their times are put together to form a team score. Their total team time is computed as X +Y. Suppose also that Mark and Raul’s times are independent. Find the mean and standard deviation of their team time.

8. A contestant on a quiz show is asked a question. If she answers the question incorrectly, then she is done and she wins no money. If she answers the question correctly, then she wins $100 and she gets to move onto another question. If she incorrectly answers the second question, then the game is over. If she correctly answers the second question, then the game is also over, but she wins an additional $500. The probability she answers the first question correctly is 0.3, and the probability she answers the second question correctly is 0.05. Assume she answers the two questions independently. Determine the probability distribution of the player’s winnings, and then find her expected winnings.