Download Math 144 tutorial 3

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