Prior to taking an essay examination, students are given 10 questions to prepare. Six of the ten will appear on the exam. One student decides to prepare only eight of the questions. Assume that the questions are equally likely to be chosen by the professor.
(a) What is the probability that they have prepared every question appearing on the test?
(b) What is the probability that both questions that they did not prepare appear on the test?
2. In the casino game of craps, a player rolls two dice and observes their sum. One of the most common bets in craps is the pass line bet. The rules of the pass line bet are as follows:
• If the sum of the dice is 7 or 11, the player wins the pass line bet. If the sum is 2, 3, or 12, the player loses.
• If the sum is a number other than 2, 3, 7, 11, or 12, a point is established and the player continues to roll. If that sum is rolled again before a 7 is rolled, the player wins. If a 7 is rolled before the sum is rolled again, then the player loses.
(a) What is the probability of a player winning the pass line bet on the first roll? What is the probability of a player losing on the first roll?
(b) Suppose the player rolls a sum of 6 on their first roll. What is the probability that they will win on the next roll? What is the probability that they will lose on the next roll?