CS2022/MA2201
HW#6

Due: Thursday, September 25

1. (3 points) Do Exercise 4.2.4 from our text.

2. (4 points) Do Exercise 4.3.6 from our text.

3. (2 points) If a group contains n men and m women, in how many ways can r people be chosen? What is the answer to this question if m=22, n=14 and r=42?

4. (3 points) Do Exercise 4.3.20 from our text. You needn’t give a numerical answer.

5. (3 points)  Jack flips one fair coin (p{heads}=p{tails}=1/2), and Jill flips two fair coins.  What is the probability that Jill obtains more heads than Jack?

6. (4 points) A deck of 10 cards, each bearing a distinct number from 1 to 10, is shuffled.  Three cards are removed, one at a time from the deck.  What is the probability that the cards are removed in sorted (increasing) order? Justify your response.

7. (7 points) (From Martin Gardner, fide Baase) Consider four fair (each face is equally likely) dice containing the following six numbers on their faces:
	D1: 1, 2, 3, 9, 10, 11
	D2: 0, 1, 7, 8, 8, 9
	D3: 5, 5, 6, 6, 7, 7
	D4: 3, 4, 4, 5, 11, 12
	• For each pair of dice Di, Dj, 1≤i,j≤4, i≠j, compute the probability in a throw of the dice, that the top face of Di will be higher than the top face of Dj. 
	• Suppose Alice and Bob are each throwing a die and whoever throws a higher number wins (nobody wins if both faces are equal). Show that if Bob chooses a die first and if Alice knows which die Bob chose, then Alice can always choose a die with a higher probability of winning than of losing.

CS2022/MA2201
HW#6 Solutions

1) Let  denote the remainder upon division by d, that is r(n)=n mod d. Since the cardinality of the codomain is d, by the Pigeonhole Principle for any d+1 elemernts in the domain must send at least two of these elements to the same image in the codomain.

2) a) 5  b) 10 c) 70 d) 1 e) 1  f) 824

3) C(m+n,r) and C(36,42)=0

4) C(40,17)=C(40,23) which is the number of ways to choose which questions are true which is the number of ways to choose which questions are false.

5) Jill gets more heads if any of the following three events occurs:
   -Jill gets 2 heads & Jack gets 0 heads
   -Jill gets 1 head and Jack gets 0 heads
   -Jill gets 2 heads and Jack gets 1 head
The three mutually exclusive events have probabilities (1/2)3, (1/2)2 and (1/2)3.  The probability of Jill getting more heads is the sum of these probabilities, or 1/2 .

6) The probability of any sequence of three distinct digits being drawn is (1/10)*(1/9)*(1/8)=1/720, so all such sequences are equiprobable.  Therefore, the experiment is equivalent to asking the probability that a random permutation (drawn from a uniform distribution) of {1,2,3} will be increasing.  Each of the six permutations being equiprobable, the answer is 1/6.

7)  The probability that Di  will beat Dj  is the entry of row i  and column j  of the array 

For any choice of die that you make (say Di ), your opponent picks a die (say Dj ) such that the corresponding entry in the array (the probability of you winning) is <1/2.  Since every row has an entry 12/36, you lose.