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psol07 - Solutions to Homework 7 Math 55 1 1 The...

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Solutions to Homework 7, Math 55 1. The probability that Beatrix starts with the crown and throws it to Andrew is 1 3 · 1 2 = 1 6 ; similarly, the probability that Charles starts with the crown and throws it to Andrew is 1 6 . Therefore, the probability that Andrew has the crown after one turn is 1 6 + 1 6 = 1 3 . Similarly, the probability that Charles has the crown after one turn is 1 3 from Andrew, and 1 3 · 1 2 = 1 6 from Beatrix, so the total probability is 1 3 + 1 6 = 1 2 . On the other hand, if Beatrix gets the crown, it must be from Charles; therefore, the probability that Beatrix has the crown after one turn is 1 3 · 1 2 = 1 6 . 2. We have p ( A ) = 4 52 = 1 13 ; p ( B ) = 13 52 = 1 4 ; and p ( C ) = 26 52 = 1 2 . Now p ( A B ) = 1 52 = p ( A ) p ( B ), and p ( A C ) = 2 52 = p ( A ) p ( C ), but p ( B C ) = 13 52 = p ( B ) p ( C ). Thus, A and B are independent, and so are A and C , but B and C are not independent. If we add a joker to the deck, then p ( A ) = 4 53 ; p ( B ) = 13 53 ; and p ( C ) = 26 53 . However, p ( A B ) = 1 53 = p ( A ) p ( B ); p ( A C ) = 2 53 = p ( A ) p ( C ); and p ( B C ) = 13 53 = p ( B ) p ( C ). Thus, in this case, no pair of events from A, B, C is independent. 3. We have p ( A B ) = p ( S - ( A B )) = 1 - p ( A B ) = 1 - ( p ( A ) + p ( B ) - p ( A B )) = 1 - p ( A ) - p ( B ) + p ( A ) p ( B ) = (1 - p ( A ))(1 - p ( B )) = p ( A ) p ( B ). Therefore, A and B are independent. Similarly, p ( A B ) = p ( A - ( A B )) = p ( A ) - p ( A B ) = p ( A ) - p ( A ) p ( B ) = p ( A )(1 - p ( B )) = p ( A ) p ( B ), so A and B are independent. (Note that in the step p ( A - ( A B )) = p ( A ) - p ( A B ), we need to use the fact that A B A .) However, if A and A are independent, then p ( A ) p ( A ) = p ( A A ) = p ( ) = 0, which implies p ( A ) = 0 or p ( A ) = 0, and in the latter case, p ( A ) = 1 - p ( A ) = 1. Thus, A and A are not independent unless p ( A ) = 0 or p ( A ) = 1. 4. Let E be the event that the number is divisible by 3, and F be the event that the number is divisible by 5. Then p ( E ) = 300 900 = 1 3 ; and p ( F ) = 180 900 = 1 5 . Also, E F is the event that the number is divisible by 15, which has probability 60 900 = 1 15 = p ( E ) p ( F ). Therefore, the two given events are independent.

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psol07 - Solutions to Homework 7 Math 55 1 1 The...

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