HW08 - University of Illinois Fall 2009 ECE 313: Problem...

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University of Illinois Fall 2009 ECE 313: Problem Set 8: Solutions Independent Events; System Reliability; CDFs 1. [“I before E except after C”] Let C = ( A B ) c denote the event that neither A nor B occurs on a trial of the experiment, and notice that one of the three events A , B , and C always occurs on a trial. Then, A occurs before B does if for n = 1 , 2 , 3 ,... , C occurs on the 1st, 2nd, . . . , ( n - 1)-th trials and A occurs on the n -th trial. By independence of the trials, P { A before B } = P ( A ) + P ( C ) P ( A ) + [ P ( C )] 2 P ( A ) + ··· = P ( A ) · 1 1 - P ( C ) = P ( A ) P ( A ) + P ( B ) . Obviously, P { B before A } = 1 - P { A before B } = P ( B ) P ( A )+ P ( B ) . The way to think about this is we can ignore all trials on which C occurs. On the very first trial on which one of A and B occurs, what are the chances that A occurs? Obviously P ( A | ( A B )) = P ( A ) P ( A B ) = P ( A ) P ( A )+ P ( B ) . See also Example 4h in Chapter 3 of Ross. 2. [What happens in Vegas stays in Vegas] (a) P (wins on first roll) = 6 36 + 2 36 = 2 9 . P (loses on first roll) = 1 36 + 2 36 + 1 36 = 1 9 . P (establishes point i on first roll) = 6 - | 7 - i | 36 ,i = 4 , 5 , 6 , 8 , 9 , 10. Check: probabilities sum to 1. (b) If the shooter’s point is i , then P (makes point | point is i ) = 6 - | 7 - i | 36 while P (craps out) = 6 36 so that P (game ends) = 12 -| 7 - i | 36 . Else, the shooter rolls the dice again. Since the events “roll i ” and “roll 7” are mutually exclusive, we can use the result of Problem 1 to get P (shooter wins | point is i ) = 6 - | 7 - i | 12 - | 7 - i | which equals 3 9 , 4 10 , 5 11 , 5 11 , 4 10
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HW08 - University of Illinois Fall 2009 ECE 313: Problem...

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