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Unformatted text preview: Solutions to Homework 02 1. We are interested in the shaded region shown in Figure 1. The shaded are on the left corresponds to A ∩ B C while that on the right is A C ∩ B . Also, the unshaded area common to both A and B is A ∩ B . Now, note that A = ( A ∩ B C ) ∪ ( A ∩ B ) ⇒ P [ A ] = P [ A ∩ B C ] + P [ A ∩ B ] ⇒ P [ A ∩ B C ] = P [ A ] P [ A ∩ B ] similarly P [ A C ∩ B ] = P [ B ] P [ A ∩ B ] Then, P [( A ∩ B C ) ∪ ( A C ∩ B )] = P [ A ∩ B C ] + P [ A C ∩ B ] = P [ A ] P [ A ∩ B ] + P [ B ] P [ A ∩ B ] = P [ A ] + P [ B ] 2 P [ A ∩ B ] A B Figure 1: Venn diagram for question 1. 2. Refer to Figure 2(a) for the region A C ∪ B C . P [ A C ∪ B C ] = 1 P [ A ∩ B ] = 1 z 1 Refer to Figure 2(b) for the region A ∩ B C . P [ A ∩ B C ] = P [ A ] P [ A ∩ B ] = x z Refer to Figure 2(c) for the region A C ∪ B . P [ A C ∪ B ] = 1 P [ A ∩ B C ] = 1 ( x z ) = 1 x + z Refer to Figure 2(d) for the region A C ∪ B C ....
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 Spring '03
 WOODS
 Set Theory, total number, unshaded

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