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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) $= \frac{5}{9},is:$
Option: 1 $\frac{4}{9}$
Option: 2 $\frac{4}{9}$
Option: 3 $\frac{4}{9}$
Option: 4 $\frac{4}{9}$
Option: 5 $\frac{1}{3}$
Option: 6 $\frac{1}{3}$
Option: 7 $\frac{1}{3}$
Option: 8 $\frac{1}{3}$
Option: 9 $\frac{5}{12}$
Option: 10 $\frac{5}{12}$
Option: 11 $\frac{5}{12}$
Option: 12 $\frac{5}{12}$
Option: 13 $\frac{2}{9}$
Option: 14 $\frac{2}{9}$
Option: 15 $\frac{2}{9}$
Option: 16 $\frac{2}{9}$

Answers (1)

best_answer

$P\left ( A\cap B \right )= P\left ( A\right )\cdot P\left ( B\right )= p\cdot 2p= 2p^{2}$

P(exactaly one of A and B)
$= P\left ( A\right )+ P\left ( B\right )-2\: \: P\left ( A\cap B \right )$
$= p+2p-4p^{2}= \frac{5}{9}\left ( given \right )$
$\Rightarrow -4p^{2}+3p= \frac{5}{9}$
$\Rightarrow -36p^{2}+27p= 5$
$\Rightarrow 36p^{2}-27p+5= 0$
$\Rightarrow 36p^{2}-12p-15p+5= 0$
$\Rightarrow 12p\left ( 3p-1 \right )-5\left ( 3p-1 \right )= 0$
$\Rightarrow p= \frac{1}{3},\frac{5}{12}$
Larger of these is $\frac{5}{12}$

Posted by

Kuldeep Maurya

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