6.042/18.062] Mathematics for Computer Science April 14, 2005 Srini devadas and Eric Lehman Problem set 9 Solutions y, Api 1 25 at 9 PM Problem 1. There are three coins: a penny, a nickel, and a quarter. When these coins are app The penny comes up heads with probability 1 3 and tails with probability 2/3 The nickel comes up heads with probability 3/4 and tails with probability 1/4 The quarter comes up heads with probability 3/5 and tails with probability 2/5 assume that the way one coin lands is unaffected by the way the other coins land. The goal of this problem is to determine the probability that an odd number of coins come up heads. For this first problem, we'll closely follow the four-step procedure for solving probability problems described in lecture. Your solution should include a tree diagram (a) What is the sample space for this experiment? Solution. We can regard each outcome as a triple indicating the orientation of the penny, nickel, and quarter. For example, the triple(, T, H)is the outcome in which the penny is heads, the nickel is tails, and the quarter is heads. The sample space is the set of all such triples: H, TI (b) What subset of the sample space is the event that an odd number of coins come up heads? Solution. The event that an odd number of coins come up heads is the subset {(H,H,H),(H,T,T),(T,H,T),(T,T,H)} (c) What is the probability of each outcome in the sample space? Solution. Edges in the tree diagram are labeled with the probabilities given in the problem statement. The probability of each outcome is the product of the probabil- ities along the corresponding root-to-leaf path. The resulting outcome probabilities are noted in the tree diagram6.042/18.062J Mathematics for Computer Science April 14, 2005 Srini Devadas and Eric Lehman Problem Set 9 Solutions Due: Monday, April 25 at 9 PM Problem 1. There are three coins: a penny, a nickel, and a quarter. When these coins are flipped: • The penny comes up heads with probability 1/3 and tails with probability 2/3. • The nickel comes up heads with probability 3/4 and tails with probability 1/4. • The quarter comes up heads with probability 3/5 and tails with probability 2/5. Assume that the way one coin lands is unaffected by the way the other coins land. The goal of this problem is to determine the probability that an odd number of coins come up heads. For this first problem, we’ll closely follow the four­step procedure for solving probability problems described in lecture. Your solution should include a tree diagram. (a) What is the sample space for this experiment? Solution. We can regard each outcome as a triple indicating the orientation of the penny, nickel, and quarter. For example, the triple (H, T, H) is the outcome in which the penny is heads, the nickel is tails, and the quarter is heads. The sample space is the set of all such triples: {H, T}3 . (b) What subset of the sample space is the event that an odd number of coins come up heads? Solution. The event that an odd number of coins come up heads is the subset: {(H, H, H),(H, T, T),(T, H, T),(T, T, H)} (c) What is the probability of each outcome in the sample space? Solution. Edges in the tree diagram are labeled with the probabilities given in the problem statement. The probability of each outcome is the product of the probabil￾ities along the corresponding root­to­leaf path. The resulting outcome probabilities are noted in the tree diagram