Assignment 4 Counting and Discrete Probability Total Points = 32 1. (4 points) (a) What is the probability that a fair coin lands Heads 5 times in a row? (b) What is the probability that a fair coin lands Heads 5 times out of 6 flips? 2. (4 points) Suppose you pick two cards, one at a time, at random, from an ordinary deck of 52 cards. Find (a) p(both cards are diamonds). (b) p(the cards have the same rank). 3. (4 points) Suppose you have 40 different books (20 math books, 15 history books, and 5 geography books). (a) You pick two books at random, one at a time. What is the probability that both books are history books? (b) You pick two books at random, one at a time. What is the probability that the two books are from different disciplines? 4. (4 points) Suppose you pick a bit string from the set of all bit strings of length 12. (a) What is the probability that the bit string has exactly two 1’s? (b) What is the probability that the bit string has exactly two 1’s, given that the string begins with a 1? 5. (4 points) Suppose that a “phrase” is any string of seven letters of the alphabet, with repeated letters allowed. (a) How many words begin with X or Y or end with X or Y? (b) How many words begin with a vowel and end with a vowel? 6. (4 points) Consider all bit strings of length 12. (a) How many begin with 110? (b) How many have exactly four 1’s? 7. (4 points) A club with 15 women and 12 men needs to form a committee of size 6. (a) How many committees are possible if the committee must have 3 women and 3 men? (b) How many committees are possible if the committee must have at least 3 women? 8. (4 points) Use the binomial theorem to expand (2c − 3d)4 . 1