Exam2___Solutions - Midterm 2 Solutions 1 A bin contains 5...

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Midterm 2 Solutions 1. A bin contains 5 red and 7 blue (distinguishable) marbles. (a) How many ways are there to pick 4 marbles out of the bin so that at least 3 are red? Solution. This can happen if either all 4 marbles are red or if 3 are red and 1 is blue. There are ( 5 4 ) ways to pick 4 red marbles out of the bin, and there are ( 5 3 ) · ( 7 1 ) ways to pick 3 red marbles and 1 blue. The total number is ( 5 4 ) + ( 5 3 ) · ( 7 1 ) = 5+10 · 7 = 75 . (b) If you pick 4 marbles out of the bin at random, what is the probability that exactly 3 are blue? Solution. There are ( 5+7 4 ) = 495 ways to pick 4 marbles out of the bin, and there are ( 5 1 ) · ( 7 3 ) = 175 ways to do so where exactly 3 marbles are blue. So the probability is 175 495 0 . 35 . (c) Suppose you repeat the following experiment 10 times: pick a marble from the bin at random, record its color, and then put the marble back in the bin. What is the probability you picked a red marble exactly 5 times out of those 10 repetitions of the experiment? Solution. Let X be the number of red marbles picked during our 10 repetitions of the experiment. Then X is a binomial random variable, with probability of success 5 5+7 = 5 12 . P X (5) = ( 10 5 ) ( 5 12 ) 5 (1 - 5 12 ) 5 = 252( 5 12 ) 5 ( 7 12 ) 5 0 . 21 .
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