Let A and B be independent events such that P(A) = p, P(B) = 2p. The largest value of p, for which P (exactly one of A, B occurs) = ${5 \over 9}$, is :
Solution
P (Exactly one of A or B)<br><br>$$ = P\left( {A \cap \overline B } \right) + \left( {\overline A \cap B} \right) = {5 \over 9}$$<br><br>$= P(A)P(\overline B ) + P(\overline A )P(B) = {5 \over 9}$<br><br>$\Rightarrow P(A)(1 - P(B)) + (1 - P(A))P(B) = {5 \over 9}$<br><br>$\Rightarrow p(1 - 2p) + (1 - p)2p = {5 \over 9}$<br><br>$\Rightarrow 36{p^2} - 27p + 5 = 0$<br><br>$\Rightarrow p = {1 \over 3}$ or ${5 \over {12}}$<br><br>${p_{\max }} = {5 \over {12}}$
About this question
Subject: Mathematics · Chapter: Probability · Topic: Classical and Axiomatic Probability
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