Homework 6

CS 280 - Spring 2002

Due: Friday, March 15

Part A

  1. Two dice are tossed.
    1. Find the probability that the sum is divisible by 5.
    2. Find the probability that the sum is divisible by 3.
    3. Find the probability that the difference is 1 (e.g., a roll of (3, 4) and a roll of (4, 3) both have a difference of 1).
    4. Find the probability that the difference is even.
       
  2. A positive integer less than 1001 is picked randomly.
    1. Find the probability that the number is divisible by 7.
    2. Find the probability that the number is divisible by 3 and 7.
    3. Find the probability that the number is divisible by 3 or 7.
    4. Find the conditional probability that the number is divisible by 3 given that it's not divisible by 7.

Part B

  1. Students had a choice of three courses: (1) Underwater Basket Weaving, (2) Applied Television Viewing, and (3) Advanced Napping.  Each student signed up for exactly one of these courses.  Applied Television Viewing (last semester's particular focus was Gilligan's Island) was twice as likely to be chosen as the other two courses, which were each equally likely.  The probability of failing each course was 0.1, 0.3, and 0.4, respectively.  Find the probability that a student took Applied Television Viewing, given that the student is known to have failed the course.
     
  2. This question assumes a standard deck of playing cards (52 cards; 13 cards each of 4 suits: clubs, diamonds, hearts, spades).  Ten cards are drawn randomly.
    1. What's the probability that exactly 5 of them are hearts?
    2. What's the probability that 5 or more of them are hearts?
    3. What's the probability that exactly 5 are hearts given that there are no spades among the 10?
    4. What's the probability that there are no spades given that exactly 5 are hearts?

Part C

  1. This question assumes a standard deck of playing cards.  A game is played in which a player draws a card. If the card is a diamond, the player receives the value of the card (i.e., $1 for an Ace, $2 for a 2, $3 for a 3, ..., $11 for a Jack, $12 for a Queen, and $13 for a King).  If the card is a heart, the player receives twice the value of the card.  If the card is black (i.e., clubs or spades) then the player loses $11.
    1. What is the expected payoff for the game?  In other words, what is the expected value of the player's winnings after playing the game once? 
    2. The game is now changed so that if an Ace is drawn (of any suit), the player wins $10.  Now what's the expected payoff for the game?
       
  2. Due to your relentless channel flipping, a new rule for TV-watching is being enforced.  You're allowed to flip once through all the channels, visiting each channel exactly once.  You can choose to stop at any point, but you cannot backup and you cannot change your mind.  Somehow, at a glance, you are able to determine a numerical rating for each TV show (even if a commercial is currently showing).  Your goal, of course, is to end up watching the show with the highest numerical rating.  For example, one strategy would be to always choose the last channel; using this strategy, the probability that you end up watching the highest rated show is 1/n where n is the number of channels.  Design a strategy (i.e., a set of simple rules) by which you are guaranteed, with probability at least 1/4, to end up at the highest-rated show  regardless of the number of channels.  Describe your strategy and show that the desired probability is 1/4 or more.  For simplicity, you can assume that the number of channels is even, that each channel is equally likely to be showing something interesting, and that no two shows have the same numerical rating.