hat is, P { X = B | Ato be executed } = 1 2 and so, P { Ato be executed | X = B } = 1 3 Similarly, P { Ato be executed | X = C } = 1 3 and thus the jailer’s reasoning is invalid. (It is true that if the jailer were to answer B, then A knows that the condemned is either himself or C, but it is twice as likely to be C .) 47. 1. 0 ≤ P ( A | B ) ≤ 1 2. P ( S | B )= P ( SB ) P ( B ) = P ( B ) P ( B ) = 1 3. For disjoint events A and D P ( A ∪ D | B )= P (( A ∪ D ) B ) P ( B ) = P ( AB ∪ DB ) P ( B ) = P ( AB )+ P ( DB ) P ( B ) = P ( A | B )+ P ( D | B ) Direct veri?cation is as follows: P ( A | BC ) P ( C | B )+ P ( A | BC c ) P ( C c | B ) = P (ABC ) P ( BC ) P ( BC ) P ( B ) + P (ABC c ) P ( BC c ) P ( BC c ) P ( B ) = P (ABC ) P ( B ) + P (ABC c ) P ( B ) = P ( AB ) P ( B ) = P ( A | B )
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