1 Easy Questions 5 points each. Three fair dice colored red, blue and green are rolled. What is the probability that at least two of them roll the same number? Three fair dice colored red, blue and green are rolled. What is the probability that at exactly two of them roll the same number? Three fair dice colored red, blue and green are rolled. What is the probability that at exactly two of them roll the same number given that the sum of the three numbers is 10? Using the axioms of probability, prove that fo any experiment and associated sample space 2, P (p) 0. In other words prove that the probability of the event corresponding to the empty set is 0 3 rabbits are playing outside their individual holes. An eagle comes and they all randomly go into a hole with one rabbit per hole. What is the probability that no rabbit went to its own hole? Assume that all configurations are equally likely You are at a casino table and the rule of the game is the following: At each step a fair coin is tossed. If it comes up H, you get 1 dollar and if it comes up tails you owe 1 dollar. What is the probability that after 100 steps you won't owe anything 8 identical chocolates are randomly divided among 3 kids. In other words, each possible way to divide is equally likely. 1. What is the probability that kid 1 get at least 3 chocolates. 2. What is the probability that kid 1 get at least 3 chocolates given that kid 2 received 3 chocolates 12 distinguishable pigeons are randomly distributed among 11 boxes with each outcome being equally likely. What is the probability that no box has more than 2 pigeons. In the hope of having a dry outdoor wedding, John and Mary decide to get married in the desert, where the average number of rainy days per year is 10. Unfortunately, the weather forecaster is predicting rain for tomorrow, the day of John and Marys wedding. Suppose that the weather forecaster is not perfectly accurate: If it rains the next day, 90% of the time the forecaster predicts rain. If it is dry the next day, 10% of the time the forecaster still incorrectly) predicts rain. Given this information, what is the probability that it will rain during John and Marys wedding? Hint: Use Bayes Theorem.l