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Math 144 tutorial 3 March 1, 2010 (Liu Zhi) Tutorial 1 / 11 1. Show that if A and B are independent, then (a). A and B0 are independent. (b). A0 and B0 are independent. (Liu Zhi) Tutorial 2 / 11 1. Show that if A and B are independent, then (a). A and B0 are independent. (b). A0 and B0 are independent. (Liu Zhi) Tutorial 2 / 11 2. We draw cards, one at a time, at random and successively from an ordinary deck of 52 cards with replacement. What is the probability that an ace appears before a face card? (Liu Zhi) Tutorial 3 / 11 2. We draw cards, one at a time, at random and successively from an ordinary deck of 52 cards with replacement. What is the probability that an ace appears before a face card? (Liu Zhi) Tutorial 3 / 11 3. A die is thrown as long as necessary if an ace or a 6 turned up. Given that no ace turned up at the first two throws, what is the probability that at least three throws will be necessary? (Liu Zhi) Tutorial 4 / 11 3. A die is thrown as long as necessary if an ace or a 6 turned up. Given that no ace turned up at the first two throws, what is the probability that at least three throws will be necessary? (Liu Zhi) Tutorial 4 / 11 4. In answering a question on a multiple-choice test, a student either knows the answer or she guesses. Let p be the probability that she know the answer and 1 − p be the probability that she guesses. Assume that a student who guesses at the answer will be correct with probability 1/m, where m is the number of multiple choice alternatives. What is the conditional probability that a student knew the answer to a question given that she answered it correctly? (Liu Zhi) Tutorial 5 / 11 (Liu Zhi) Tutorial 6 / 11 5. In an urn are 5 fair coins, two 2-headed coins, and four 2-tailed coins. A coin is to be randomly selected and flipped. Compute the probability that the coin is fair if the result was (a). The flip was head; (b). 2 flips were both heads. Use prior and posterior probabilities. (Liu Zhi) Tutorial 7 / 11 5. In an urn are 5 fair coins, two 2-headed coins, and four 2-tailed coins. A coin is to be randomly selected and flipped. Compute the probability that the coin is fair if the result was (a). The flip was head; (b). 2 flips were both heads. Use prior and posterior probabilities. (Liu Zhi) Tutorial 7 / 11 (Liu Zhi) Tutorial 8 / 11 (Liu Zhi) Tutorial 9 / 11 6. A random variable X is called symmetric if for all real number x such that P(X ≥ x) = P(X ≤ −x). Prove that if X is symmetric, then for all t > 0, its distribution function F satisfies the following relations: (a). P(|X| ≤ t) = 2F(t) − 1, (b). P(|X| > t) = 2(1 − F(t)), (c). P(X = t) = F(t) + F(−t) − 1. (Liu Zhi) Tutorial 10 / 11 6. A random variable X is called symmetric if for all real number x such that P(X ≥ x) = P(X ≤ −x). Prove that if X is symmetric, then for all t > 0, its distribution function F satisfies the following relations: (a). P(|X| ≤ t) = 2F(t) − 1, (b). P(|X| > t) = 2(1 − F(t)), (c). P(X = t) = F(t) + F(−t) − 1. (Liu Zhi) Tutorial 10 / 11 (Liu Zhi) Tutorial 11 / 11