CISC5400 Discrete Structures

Spring 2017

 

Assignment 4

Due May 11

 

 

1.      Find a close formula for the sequence 1, 2, 4, 7, 11, 16, ….

 

2.      Find a close formula for the sequence 7, 13, 25, 43, 67, …

 

3.      Find a close formula for the sequence 1, 3, 8, 16, 27, 41, …

 

4.      Find a close formula for the sequence 2, 7, 17, 37, 77, …

 

5.      Find a close formula for the sequence 1, 5, 17, 53, 161, …

 

6.      If one card is to be drawn from a standard deck of cards, find the probability that (a) The card is a red Ace. (b) The card is either red or an ace.

 

7.      A club with 12 female members and 11 male members is forming a budget committee consisting of three club members. Assuming each committee is equally likely to be appointed, find that probability that (a) The committee has at least one female. (b) David and Jenny are not both on the committee.

   

8.      A card is drawn from an ordinary deck of 52 cards. Show how to use the general sum rule to find the probability that the card is (a) An ace or a black card. (b) A diamond, a club, or a king.

 

9.      Two cards are drawn from a deck, with replacement. Show how to use the product rule to find the probability that (a) Both cards are spades. (b) At least one card is an ace. (Hint: Consider the complementary problem.)

 

10.  What is the probability that when a pair of dice is rolled, either (at least) one die shows 5 or the dice sum to 8?

 

11.  In certain poker games, five-card hands consist of three face-down cards and two face-up cards. In particular, you can see two of the cards in your opponent’s hand. For each question below, specify events E1 and E2 so that the given question is about Prob(E2|E1), and then answer the probability question. (a) What is the probability that his hand is a flush, given that you can see he has the ace and 5 of clubs? (b) What is the probability that his hand is a flush, given that you can see he has the ace and 5 of clubs, and you know your own hand consists of all clubs?

 

 

12.  (a) What is the probability that a fair coin will come up heads exactly three times in six tosses? (b) What is the probability that a fair die will come up as a 5 or 6 exactly twice in four tosses?

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