test1 - MATH 301/601, Test 1, February 8, 2006 1. From the...

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Unformatted text preview: MATH 301/601, Test 1, February 8, 2006 1. From the integers 1,...,10, three numbers are chosen at random without replacement and disregarding the order. (a) In how many ways can this be done? (b) What is the probability that the smallest number is 4? (c) Do (a) and (b) with 10 replaced by an integer n and 4 replaced by an integer k (where n 3 and k n - 2) 2. Consider two urns such that urn I has two black balls and urn II has one black ball and one white ball. You choose an urn at random and then pick a ball from this urn. (a) If the ball is black, what is the probability that you chose urn I? (b) If you draw twice with replacement from the chosen urn and get two black balls, what is the probability that you chose urn I? (c) If you draw j times with replacement from the chosen urn and only get black balls, what is the probability that you chose urn I? 3. Let A and B be events. Are the following statements true or false? (a) It is possible to have P (A B) > P (A) (b) It is possible to have P (A B) = P (A) (c) It is possible to have P (B) > 0 and P (A B) = P (A) (d) It is always true that P (A) P (A B) (e) It is always true that P (A|B) + P (A|B c ) = 1 (f ) It is always true that P (A|B) + P (Ac |B) = 1 (g) It is always true that P (A) = P (A B) + P (A B c ) (h) If A and B are independent, then Ac and B c are independent (i) If A and B are independent, then P (A B) = P (A) + P (B) (j) If A and B are independent, then P (A B) = P (A) + P (B)P (Ac ) 4. If you are correct with probability p on each question in problem 3, what is the probability to get at least 9 correct answers? Assume independence between questions. ...
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This note was uploaded on 04/08/2008 for the course MATH 301 taught by Professor Staff during the Spring '08 term at Tulane.

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