Verified Solution • 2025–26 Session moderate

QJEEM21Mae202: JEE Main 2021 Mathematics - Class 12 JEEM21Mae202 Solution: Let A and B be independent events such that P(A) = p, P(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) = 59{5 \over 9} , is :

Multiple Choice Options

  • A 13{1 \over 3}
  • B 29{2 \over 9}
  • C 49{4 \over 9}
  • D 512{5 \over 12}

Step-by-Step Verified Solution

P (Exactly one of A or B)

=P(A∩B‾)+(A‾∩B)=59= P\left( {A \cap \overline B } \right) + \left( {\overline A \cap B} \right) = {5 \over 9}

=P(A)P(B‾)+P(A‾)P(B)=59= P(A)P(\overline B ) + P(\overline A )P(B) = {5 \over 9}

⇒P(A)(1−P(B))+(1−P(A))P(B)=59\Rightarrow P(A)(1 - P(B)) + (1 - P(A))P(B) = {5 \over 9}

⇒p(1−2p)+(1−p)2p=59\Rightarrow p(1 - 2p) + (1 - p)2p = {5 \over 9}

⇒36p2−27p+5=0\Rightarrow 36{p^2} - 27p + 5 = 0

⇒p=13\Rightarrow p = {1 \over 3} or 512{5 \over {12}}

pmax⁡=512{p_{\max }} = {5 \over {12}}

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