Question

A black and a red die are rolled together. Find the conditional probability of obtaining the sum $8$, given that the red die resulted in a number less than $4$.

Answer 1

Let $A$: getting a total of $8$
$A=(2,6),(3,5),(4,4),(5,3),(6,2)$
= $5$ elements
$B$: Red die resulted in a number less than $4$
= $(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)$
$=18$ elements
$\therefore A\cap B=(2,6),(3,5)=2$ elements
Now,
$P\left(A|B\right)=\frac{P(A\cap B)}{P\left(B\right)}=\frac{\frac{2}{36}}{\frac{18}{36}}=\frac{2}{18}=\frac{1}{9}$.