Question

If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one of Aand B occurs) =  \[\frac{5}{9}\], then find the value of p.

Answer 1

\[\text{ As, A and B are independent events .}  \]
\[\text{ So } , P\left( A \cap B \right) = P\left( A \right) \times P\left( B \right) . . . . . \left( i \right)\]
\[\text{ Now } , \]
\[P\left( \text{ Exactly one of A and B occurs } \right) = \frac{5}{9}\]
\[ \Rightarrow P\left( \text{ Only }  A \right) + P\left( \text{ Only }B \right) = \frac{5}{9}\]
\[ \Rightarrow \left[ P\left( A \right) - P\left( A \cap B \right) \right] + \left[ P\left( B \right) - P\left( A \cap B \right) \right] = \frac{5}{9}\]
\[ \Rightarrow \left[ P\left( A \right) - P\left( A \right) \times P\left( B \right) \right] + \left[ P\left( B \right) - P\left( A \right) \times P\left( B \right) \right] = \frac{5}{9} \left[ \text{ Using }  \left( i \right) \right]\]
\[ \Rightarrow P\left( A \right) \times \left[ 1 - P\left( B \right) \right] + P\left( B \right) \times \left[ 1 - P\left( A \right) \right] = \frac{5}{9}\]
\[ \Rightarrow p\left[ 1 - 2p \right] + 2p\left[ 1 - p \right] = \frac{5}{9}\]
\[ \Rightarrow p - 2 p^2 + 2p - 2 p^2 = \frac{5}{9}\]
\[ \Rightarrow 3p - 4 p^2 = \frac{5}{9}\]
\[ \Rightarrow 27p - 36 p^2 = 5\]
\[ \Rightarrow 36 p^2 - 27p + 5 = 0\]
\[\text{ So, using quadratic formula, we get } \]
\[p = \frac{- \left( - 27 \right) \pm \sqrt{\left( - 27 \right)^2 - 4 \times 36 \times 5}}{2 \times 36}\]
\[ = \frac{27 \pm \sqrt{729 - 720}}{72}\]
\[ = \frac{27 \pm \sqrt{9}}{72}\]
\[ = \frac{27 \pm 3}{72}\]
\[ \Rightarrow p = \frac{27 + 3}{72} or p = \frac{27 - 3}{72}\]
\[ \Rightarrow p = \frac{30}{72} or p = \frac{24}{72}\]
\[ \therefore p = \frac{5}{12} or p = \frac{1}{3}\]