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Unformatted text preview: MATH 301/601, Solutions to test 1 1(a). From the integers 1,...,10, three numbers are chosen at random with- out 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) Solution (a) 10 3 = 120 (b) There are 6 2 = 15 ways to choose the three numbers so that the small- est is 4: the number 4 must be chosen and the remaining two numbers can be chosen among the six numbers 5 to 10. The probability is therefore 15 / 120 = 1 / 8. (c) n 3 and n- k 2 / n 3 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?black balls, what is the probability that you chose urn I?...
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