Let A and B be independent events such that P(A) = p, P(B) = 2p.

Question

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 :

Accepted Answer

P (Exactly one of A or B)

$$ = P\left( {A \cap \overline B } \right) + \left( {\overline A  \cap B} \right) = {5 \over 9}$$

$$ = P(A)P(\overline B ) + P(\overline A )P(B) = {5 \over 9}$$

$$ \Rightarrow P(A)(1 - P(B)) + (1 - P(A))P(B) = {5 \over 9}$$

$$ \Rightarrow p(1 - 2p) + (1 - p)2p = {5 \over 9}$$

$$ \Rightarrow 36{p^2} - 27p + 5 = 0$$

$$ \Rightarrow p = {1 \over 3}$$ or $${5 \over {12}}$$

$${p_{\max }} = {5 \over {12}}$$

Solutions · 1