$A$ and $B$ are two independent events. The probability that both $A$ and $B$ occur is $\frac{1}{6}$

Question

$A$ and $B$ are two independent events. The probability that both $A$ and $B$ occur is $\frac{1}{6}$ and probability that neither of them occur is $1 / 3$. Find the possible probabilities of the occurrence of $A$.

Professor PC Thomas Classes · Answer

Let $P(A)=a$ and $P(B)=b$ Given $P(A) P(B)=\frac{1}{6} \Rightarrow a b=\frac{1}{6}$\ Also $P(\bar{A}) P(\bar{B})=\frac{1}{3}$\ $\Rightarrow(1-P(A))(1-P(B))=\frac{1}{3}$\ $\Rightarrow 1-P(A)-P(B)+P(A) P(B)=\frac{1}{3}$\ $\Rightarrow 1-a-b+a b=\frac{1}{3} \quad$ As $a b=\frac{1}{6}, a+b=\frac{5}{6}$\ Solving $a=\frac{1}{3}, b=\frac{1}{2}$ or $a=\frac{1}{2}, b=\frac{1}{3}$\ $	herefore P(A)=\frac{1}{3}$ or $P(A)=\frac{1}{2}$.\