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2 using the results of a or otherwise nd the value of

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Unformatted text preview: (A ∩ B ) = 1 − 0.1 = 0.9 Chan Chi Ho, Chan Tsz Hin & Shi Yun Example class 2 Problem 2 Problem 2 A and B are two events. Suppose that P (A|B ) = 0.6, P (B |A) = 0.3 and P (A ∪ B ) = 0.72. LetP (A) = a. 1 Express P (A ∩ B ) and P (B ) in terms of a. 2 Using the results of (a), or otherwise, find the value of a. 3 Are A and B independent events? Explain your answer briefly. Chan Chi Ho, Chan Tsz Hin & Shi Yun Example class 2 Solution to Problem 2 Solution: 1 P (A ∩ B ) = P (B |A)P (A) = 0.3a P (A|B )P (B ) = P (A ∩ B ) 0.6P (B ) = 0.3a P (B ) = 0.5a 2 P (A ∪ B ) = P (A) + P (B ) − P (A ∩ B ) 0.72 = a + 0.5a − 0.3a a = 0.6 3 P (A|B ) = 0.6 = P (A) and P (B |A) = 0.3 = 0.5 × 0.6 = P (B ) therefore A and B are independent events. Chan Chi Ho, Chan Tsz Hin & Shi Yun Example class 2 Problem 3 Problem 3 A and B are two events. Suppose that P (B c |A) = 3 , P (Ac |B ) = 3 and 4 5 2 P (Ac ) = 5 , where Ac and B c are complementary events of A and B respectively. Let P (B ) = p , where 0 < p < 1. 1 Find P (A ∩ B c ). 2 Express P (Ac ∩ B ) in terms of p . 3 Using the fact that Ac ∪ B is the complementary event of A ∩ B c , or otherwise, find the value of p . 4 Are A and B mutually exclusive? Explain your answ...
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