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# hw2 - 5 There is a weather forecast in the morning If a...

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Homework # 2, Due Tuesday, January 24 1. Problem. a) We toss a fair coin n times. Let a n be the number of outcomes which do not contain a sequence ...HHH . .. (that is, do not contain 3 or more heads in a row). Find a recurrence formula relating numbers a n . b) We toss a fair coin 10 times. What is the probability that there will be 3 or more heads in a row? 2. Problem. There are 3 coins in a box, one of which is double-headed, one is double- tailed and one is a fair coin. a) We pick a coin at random and toss it. What is the probability that the coin lands heads? b) We see that the coin indeed landed heads. What is the probability that it is the double-headed coin? c) We toss the same coin again. What is the probability that it lands heads? d) We see that the coin indeed landed heads. What is the probability that it is the double-headed coin? 3. Problem. In London, every day is either rainy with probability 3 / 5 or dry with probability 2
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Unformatted text preview: / 5. There is a weather forecast in the morning. If a rainy day is forecasted, it rains with probability 3 / 4 and if a dry day is forecasted, it is dry with probability 2 / 3. Mr. Pickwick listens to the forecast. If a rainy day is forecasted, he takes his umbrella before going out. If a dry day is forecasted, he takes his umbrella with probability 1 / 3. What is the probability that the day is dry, given that Mr. Pickwick took his umbrella? 4. Problem. There are three roads from city A to city B and there are two roads from city B to city C . Each of the ±ve roads is blocked with probability p , independently of the others. What is the probability that one can travel from city A to B given that there is no open route from A to C ? 5. Problem. We toss a fair coin till the coin lands 10 heads in total. What is the probability that we make 20 tosses altogether? 1...
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