Probability Homework Problem 1 We draw the top 7 cards from a well-shuffled standard 52-card deck. Find the probability that: (a) The 7 cards include exactly 3 aces. (b) The 7 cards include exactly 2 kings. (c) The probability that the 7 cards include exactly 3 aces. or exactly 2 kings, or both.

Problem 2 Alice and Bob have 2n+1 coins, each coin with probability of heads equal to 1/2. Bob tosses n+1 coins, while Alice tosses the remaining n coins. Assuming independent coin tosses, show that the probability that after all coins have been tossed, Bob will have gotten more heads than Alice is 1/2.

Problem 3 We are given three coins: one has heads in both faces, the second has tails in both faces, and the third has a head in one face and a tail in the other. We choose a coin at random, toss it, and the result is heads. What is the probability that the opposite face is tails?

Problem 4 Each of k jars contain m white and n black balls. A ball is randomly chosen from jar 1 and transferred to jar 2, then a ball is randomly chosen from jar 2 and transferred to jar 3, etc. Finally a ball is randomly chosen from jar k. Show that the probability that the last ball is white is the same as probability that the first ball is white, i.e. it is m/(m+n).

Problem 5 A power utility can supply electricity to a city from n different power plants. Power plant i fails with probability pi, independent of the others. (a) Suppose that any one plant can produce enough electricity to supply the entire city. What is the probability that the city will experience a black-out? (b) Suppose that two power plants are necessary to keep the city from a black- out. Find the probability that the city will experience a black-out.