Question

Let $A$ and $B$ be two independent events such that $P\left(A\right)=\frac{1}{3}$ and $P\left(B\right)=\frac{1}{6}$. Then, which of the following is TRUE ?

• A. $P\left(A|\right(A\cup B\left)\right)=\frac{1}{4}$
• B. $P\left(A|B^{\prime }\right)=\frac{1}{3}$
• C. $P\left(A|B\right)=\frac{2}{3}$
• D. $P\left(A^{\prime }|B^{\prime }\right)=\frac{1}{3}$

Answer 1

The correct option is B $P\left(A|B^{\prime }\right)=\frac{1}{3}$

If $X$ and $Y$ are independent events,
then
$P(X\cap Y)=P\left(X\right)P\left(Y\right)$ and
$X$ and $Y^{\prime }$ are also independent.
So, $P\left(X|Y\right)=\frac{P(X\cap Y)}{P\left(Y\right)}=\frac{P\left(X\right)P\left(Y\right)}{P\left(Y\right)}=P\left(X\right)$
Therefore, $P\left(A|B\right)=P\left(A\right)=\frac{1}{3}$
and $P\left(A|B^{\prime }\right)=P\left(A\right)=\frac{1}{3}$