Question

A fair die is rolled. Consider events $E=\{1,3,5\},F=\{2,3\}$ and $G=\{2,3,4,5\}$
Then the value of $P\left(\right(E\cup F\left)/G\right)$ is:

• A. $\frac{1}{3}$
• B. $\frac{1}{4}$
• C. $\frac{2}{3}$
• D. $\frac{3}{4}$

Answer 1

The correct option is D $\frac{3}{4}$

The sample space for the given experiment is $S=\{1,2,3,4,5,6\}$
Here, $E=\{1,3,5\},F=\{2,3\}$ and $G=\{2,3,4,5\}$

$E\cup F=\{1,2,3,5\}$
$\Rightarrow (E\cup F)\cap G=\{2,3,5\}$
$\Rightarrow P\left(\right(E\cup F)\cap G)=\frac{3}{6}=\frac{1}{2}$
Also $P\left(G\right)=\frac{4}{6}=\frac{2}{3}$

Now, we know that
By definition of conditional probability,
$P\left(E/F\right)=\frac{P(E\cap F)}{P\left(F\right)}$
$\Rightarrow P\left(\right(E\cup F\left)/G\right)=\frac{P\left(\right(E\cup F)\cap G)}{P\left(G\right)}=\frac{1/2}{2/3}=\frac{3}{4}$