Problem 1. The two-headed coin. A box contains three coins: two regular coins and one fake...

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Problem 1. The two-headed coin. A box contains three coins: two regular coins and one fake two-headed coin. (a) You pick a coin at random and toss it. What is the probability that it lands heads up? (b) You pick a coin at random and toss it, and get heads. What is the probability that it is the two-headed coin? Problem 2. Rain rain go away! In my town, it's rainy one third of the days. Given that it is rainy, there will be heavy traffic with probability 0.5, and given that it is not rainy, there will be heavy traffic with probability 0.25. If it's rainy and there is heavy traffic, I arrive late for work with probability 0.5. On the other hand, the probability of being late is reduced to 0.125 if it is not rainy and there is no heavy traffic. In other situations (rainy and no traffic, not rainy and traffic) the probability of being late is 0.25. You pick a random day. (a) What is the probability that it's not raining and there is heavy traffic and I am not late? (b) What is the probability that I am late? (c) Given that I arrived late at work, what is the probability that it rained that day? Problem 1. The two-headed coin. A box contains three coins: two regular coins and one fake two-headed coin. (a) You pick a coin at random and toss it. What is the probability that it lands heads up? (b) You pick a coin at random and toss it, and get heads. What is the probability that it is the two-headed coin? Problem 2. Rain rain go away! In my town, it's rainy one third of the days. Given that it is rainy, there will be heavy traffic with probability 0.5, and given that it is not rainy, there will be heavy traffic with probability 0.25. If it's rainy and there is heavy traffic, I arrive late for work with probability 0.5. On the other hand, the probability of being late is reduced to 0.125 if it is not rainy and there is no heavy traffic. In other situations (rainy and no traffic, not rainy and traffic) the probability of being late is 0.25. You pick a random day. (a) What is the probability that it's not raining and there is heavy traffic and I am not late? (b) What is the probability that I am late? (c) Given that I arrived late at work, what is the probability that it rained that day?