UMBC CMSC203 Discrete Structures, Section 06, Spring 2016

Homework 11, Due Thursday, 04/28

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

1. Marble Placement. In a board game, you have 19 indistinguishable marbles that you can place in 5 distinguishable locations. You must place at least 2 marbles at each location, but are otherwise allowed to place as many or as few marbles at each location. How many different ways can you make these placements?

2. Balls & Bins. You have 13 balls that you throw at 5 bins labeled A, B, C, D and E. Our assumption is that when a ball is thrown at the bins, there is an equal probability that the ball lands in any particular bin. Also, the ball will always land in one of the bins. Each bin is large enough to hold any number of balls.
   1. You throw the 13 balls, one at a time, at the bins. What is the probability that exactly 3 balls land in bin A? Justify your answer.
   2. You throw the 13 balls, one at a time, at the bins. What is the probability that 4 or fewer balls land in bin B? Justify your answer.

3. Two Urns. [Adapted from Epp, 3/e.]
   You have two urns. One urn holds 5 red balls and 13 yellow balls. The second urn holds 9 red balls and 11 yellow balls. You pick one ball using this procedure: randomly pick one of the two urns with equal probability, then pick a ball from the chosen urn so that each ball is chosen with equal probability.
   1. What is the probability that the chosen ball is red?
   2. If the chosen ball is red, what is the probability that the chosen ball came from the first urn?

4. Odd Man Out. Four friends play a game called Odd Man Out. They each flip a fair coin. If 1 person has heads and the other 3 have tails, then the person with heads is the odd man. Similarly, if 1 person has tails and the other 3 have heads, then the person with tails is the odd man. What is the probability of having an odd man after each person flips just once? Explain your answer.

5. Plastic Utensils. You randomly pick utensils from a box with plastic knives, forks and spoons. Each time you pick, there is an equal probability of picking any of the utensils remaining in the box. Initially, the box holds 4 forks, 3 spoons and 7 knives.
   Note: Show all of your work and explain your answers.

   1. Suppose you pick 3 utensils without replacement. What is the probability that you picked a fork, a spoon and a knife (in any order)?

   2. Suppose that you pick 2 utensils without replacement. What is the probability that the second utensil you picked is a knife?

   3. Suppose that you pick 2 utensils without replacement. What is the conditional probability that the second utensil you picked is a knife given that the first utensil is a fork?

   4. Suppose that you pick 2 utensils without replacement. What is the probability that at least one of the two is a spoon?