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HW6_sol - CSE21 FA11 Homework#6 6.1 Five boys and three...

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CSE21 FA11 Homework #6 (10/31/11) 6.1. Five boys and three girls are throwing Frisbees. Each boy has one Frisbee and throws it to a random girl. What is the probability that each of the girls gets at least one of thrown Frisbees? Answer : S (5 , 3) * ( 3! 3 5 ) = 25 × 6 243 = 0 . 6172. 6.2. A box contains 20 light bulbs, 5 of which are defective (and the rest are good). You randomly select 8. What is the expected number of good bulbs you get this way? Answer : There are two ways to do this problem. The straightforward is to grind it out using the definition of expected value. The probability of drawing k good bulbs for 3 k 8 is ( 15 k )( 5 8 - k ) ( 20 8 ) Hence, the expected number of good bulbs drawn is 8 X k =3 k ( 15 k )( 5 8 - k ) ( 20 8 ) = 6 . An easier way is to use linearity of expectation. We can imagine the bulbs are drawn out sequentially without replacement as ( B 1 , B 2 , . . . , B 8 ). As usual, define the random variable X ( B 1 , . . . , B 8 ) to be the number of good bulbs among the B i , and decompose X = X 1 + X 2 + . . . + X 8 , where X i ( B 1 , . . . , B 8 ) = 1 if B i is good, and 0 otherwise. Since the expected value
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