UMBC CMSC203 Discrete Structures, Section 06, Spring 2016

Homework 12, Due Thursday, 05/05

When factorials are involved, leave your answer in terms of factorials (e.g., 5!/(3! ⋅ 2)).

1. 4- and 8-sided Die. You have a fair 4-sided die and a fair 8-sided die. (Here a die is fair if when the die is rolled, there is an equal probability for any particular side to be on top.) The sides of the 4-sided die are labeled with 2, 7, 7 and 9. The sides of the 8-sided die are labeled with 1, 1, 2, 3, 3, 4, 5 and 5.
   1. What is the expected value of the number on top when you roll the 4-sided die?
   2. What is the expected value of the number on top when you roll the 8-sided die?
   3. What is the expected value of the sum of the numbers on top when you roll both dice?

2. Rolling for 10+. You have two normal 6-sided dice (with sides labeled 1, 2, 3, 4, 5 and 6). Assume that the dice are fair. Suppose that you keep rolling the two dice until the sum of the numbers on the dice is greater than or equal to 10. What is the expected number times that you will roll the dice?

3. Two Aces. You select 6 cards from a standard deck of playing cards. Of the 6 cards, 2 are Aces and 4 are not. You shuffle the cards well (so each permutation of the 6 cards is equally likely). Then, you deal out the selected cards (without replacement) face up until two Aces are shown. What is the expected number of cards dealt?

   Hint: To calculate the probability that exactly 4 cards are dealt until two Aces are shown, consider the fact that the first Ace could be the first card, the second card or the third card. The fourth card is, of course, the second Ace. Then no more cards are dealt.

4. Missing Socks. You have 7 pairs of socks, all different. Let's call them red, orange, yellow, green, blue, purple and violet (because indigo shouldn't be the name of a color.) You put these 14 individual socks in the wash. As you suspected, the dryer does indeed select socks with equal probability and eats them. When the wash is done you only have 11 socks.
   1. What is the probability that you have both red socks?
   2. What is the expected number of pairs of socks that you get back? (You need to have both socks of the same color to count as a pair.) Hint: use linearity of expectations.