HW11Answers(3.5) - i th missile (with i ∈ { 1 , 2 , 3 }...

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Math331, Spring 2008 Instructor: David Anderson Section 3.5 Homework Answers Homework: pg. 119 #’s 2, 4, 15, 17, 29 2. No, these events are not independent. Because they are old friends, we may assume that P ( A ) > 0 and P ( B ) > 0. However, P ( AB ) = 0 n = P ( A ) P ( B ). 4. Note that for any Frst role, the probability of the sum being odd is 1/2. (for example, if the Frst role is a 1, there is a probability of .5 that the sum will be odd). Using the law of total probability then shows that if A is the event that the sum of the outcomes is odd, then P ( A ) = 1 / 2. Clearly P ( B ) = 1 / 6. Also, P ( AB ) = 3 / 36 = 1 / 12 = (1 / 2)(1 / 6) = P ( A ) P ( B ). Therefore, the events are independent. 15. We suppose A and B are independent with A B . Then, P ( AB ) = P ( A ) P ( B ) by independence. But because A B , we have that P ( AB ) = P ( A ). Combining the previous two facts gives us P ( A ) = P ( A ) P ( B ). If P ( A ) n = 0, then we can divide by P ( A ) and see P ( B ) = 1. Therefore, either P ( A ) = 0 or P ( B ) = 1 (or both). 17. Let H i be the event that the target is hit with the
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Unformatted text preview: i th missile (with i ∈ { 1 , 2 , 3 } ). Then, by independence, the probability of a hit is P ( H 1 ∪ H 2 ∪ H 3 ) = P ( H 1 ) + P ( H 2 ) + P ( H 3 ) − P ( H 1 H 2 ) − P ( H 1 H 3 ) − P ( H 2 H 3 ) + P ( H 1 H 2 H 3 ) = . 7 + . 8 + . 9 − P ( H 1 ) P ( H 2 ) − P ( H 1 ) P ( H 3 ) − P ( H 2 ) P ( H 3 ) + P ( H 1 ) P ( H 2 ) P ( H 3 ) = 2 . 4 − . 56 − . 63 − . 72 + . 504 = . 994 . 29. Let E i be the event that the i th switch is closed. Then, for a signal to get through, we need one of the following events to occur: E 1 E 2 E 4 E 6 or E 1 E 3 E 5 E 6 . By the independence of the switches we have P ( E 1 E 2 E 4 E 6 ∪ E 1 E 3 E 5 E 6 ) = P ( E 1 E 2 E 4 E 6 ) + P ( E 1 E 3 E 5 E 6 ) − P ( E 1 ··· E 6 ) = 2 p 4 − p 6 . 1...
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This note was uploaded on 03/31/2008 for the course MATH 331 taught by Professor Anderson during the Spring '08 term at Wisconsin.

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