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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) <b

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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