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Unformatted text preview: AMS 410 Actuarial Mathematics Fall 2009 Homework 1 Solution: General Probability 09/15/2009 1. Given that P [ A B ] = 0 . 7 and P [ A B ] = 0 . 9 , nd P [ A ] . Using Venn Diagram (the region denotes the probability of the corresponding event), P [ A B ] = N 1 + N 3 + N 4 = 0 . 9 , thus N 2 = 0 . 1 . P [ A B ] = N 1 + N 2 + N 3 = 0 . 7 , so N 1 + N 3 = . 7 . 1 = 0 . 6 = P [ A ] . Note: When you have two factors /events, use the Venn Diagram or the box . 2. Given that A and B are independent with P [ A ] = 2 P [ B ] and P [ A B ] = 0 . 15 , nd P [ A B ] . By independence, P [ A B ] = P [ A ] P [ B ] = 2 P [ B ] P [ B ] = 2 P [ B ] 2 = 0 . 15 . Thus P [ B ] = . 274 , P [ B ] = 0 . 726 ; P [ A ] = 0 . 548 , P [ A ] = 0 . 452 . Since A and B are also independent, P [ A B ] = P [ A ] P [ B ] = 0 . 328 . 3. Given that A and B are independent with P [ A B ] = 0 . 8 and P [ B ] = 0 . 3 , nd P [ A ] . Given P [ B ] = 0 . 3 , P [ B ] = 0 . 7 . By independence, P [ A B ] = P [ A ] + P [ B ] P [ A ] P [ B ] = P [ A ] + 0 . 7 . 7 P [ A ] = 0 . 7 + 0 . 3 P [ A ] = 0 . 8 , thus P [ A ] = 1 3 = 0 . 333 . 4. Given that P [ A ] = 0 . 2 , P [ B ] = 0 . 7 and P [ A  B ] = 0 . 15 , nd P [ A B ] . P [ A B ] = P [ B ] P [ A  B ] = 0 . 7 . 15 = 0 . 105 . P [ A B ] = P [( A B ) ] = 1 P [ A B ] = 1 ( P [ A ] + P [ B ] P [ A B ]) = 1 (0 . 2 + 0 . 7 . 105) = 0 . 205 . 5. Assuming A , B and C are mutually independent, with P [ A ] = P [ B ] = P [ C ] = 0 . 1 , compute: (a) P [ A B ] By independence, P [ A B ] = P [ A ] P [ B ] = 0 . 01 . P [ A B ] = P [ A ] + P [ B ] P [ A B ] = . 1 + 0 . 1 . 01 = 0 . 19 . (b) P [ A B C ] By independence, P [ A B ] = P [ A ] P [ B ] = 0 . 01 = P [ A C ] = P [ B C ] , P [ A B C ] = P [ A ] P [ B ] P [ C ] = 0 . 001 . P [ A B C ] = P [ A ] + P [ B ] + P [ C ] P [ A B ] P [ A C ] P [ B C ] + P [ A B C ] = 0 . 3 . 03 + 0 . 001 = 0 . 271 ....
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This note was uploaded on 08/08/2010 for the course AMS 410 taught by Professor Yang,y during the Fall '08 term at SUNY Stony Brook.
 Fall '08
 Yang,Y

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