# HE2_Fall97 - University Solutions to Second Hour Exam ECE...

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University Solutions to Second Hour Exam ECE 313 of Illinois Fall 1997 1. An experiment consists of tossing a fair coin repeatedly until a Head occurs for the tenth time. (a) Suppose that the tenth Head occurred on the twentyfifth toss. What is the conditional probability that the first toss resulted in a Head? (b) Let the random variable X denote the number of Tails that occur before the tenth Head occurs. For example, if the tenth Head occurred on the twentyfifth toss as was assumed in part (a), then X = 15 for this particular outcome of the experiment. More generally, X can take on any non-negative integer value 0, 1, 2, 3, … . What is P{ X = k}? 1.(a) P{first toss is a Head | tenth Head on twentyfifth toss} = P{first toss is a Head tenth Head on twentyfifth toss} P{tenth Head on twentyfifth toss} = P{H; 8 Heads in 23 tosses; H} P{tenth Head on twentyfifth toss} = () 1 2 23 8 1 2 8 1 2 23–8 1 2 24 9 1 2 25 = 23 8 24 9 = 23!9!15! 24!8!15! = 9 24 = 3 8 . Alternatively, given that the tenth Head was on the 25th toss, we know for sure that nine Heads occurred on 24 tosses. Thus, the probability that the first toss resulted in a Head is 9/24 according to a formula derived in class.

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HE2_Fall97 - University Solutions to Second Hour Exam ECE...

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