Question

A and B are two independent events. The probability that A and B occur is 1/6 and the probability that neither of them occurs is 1/3. Find the probability of occurrence of two events.

Answer 1

\[P\left( A \cap B \right) = P\left( A \right) P\left( B \right) \left( \text{ A and B are independent events }  \right)\]
\[\frac{1}{6} = P\left( A \right) P\left( B \right)\]
\[ \Rightarrow P\left( A \right) = \frac{1}{6P\left( B \right)} . . . \left( 1 \right)\]
\[P\left( \bar{A} \cap \bar{B} \right) = \left[ 1 - P\left( A \right) \right]\left[ 1 - P\left( B \right) \right]\]
\[ \Rightarrow \frac{1}{3} = \left[ 1 - P\left( A \right) \right]\left[ 1 - P\left( B \right) \right]\]
\[ \Rightarrow \frac{1}{3} = \left[ 1 - \frac{1}{6P\left( B \right)} \right]\left[ 1 - P\left( B \right) \right] \left[ \text{ Using }  \left( 1 \right) \right]\]
\[\text{ Let }  P\left( B \right) = x\]
\[ \Rightarrow \left( \frac{6x - 1}{6x} \right)\left( 1 - x \right) = \frac{1}{3}\]
\[ \Rightarrow 6x - 1 - 6 x^2 + x = 2x\]
\[ \Rightarrow 6 x^2 - 5x + 1 = 0\]
\[ \Rightarrow \left( 2x - 1 \right)\left( 3x - 1 \right) = 0\]
\[ \Rightarrow x = \frac{1}{2} or x = \frac{1}{3}\]
\[\text { If } P\left( B \right) = \frac{1}{2}, \text{ then}  P\left( A \right) = \frac{1}{3}\]
\[\text { If  }P\left( B \right) = \frac{1}{3}, \text{ then }  P\left( A \right) = \frac{1}{2}\]