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Q.
A dice is rolled twice and the sum of the numbers appearing on them is observed to be $7$. What is the conditional probability that the number $2$ has appeared at least once ?
Let $A$ and $B$ be two events such that
$A=$ getting number 2 at least once
$B=$ getting 7 as the sum of the numbers on two dice
Here,
$A=\{(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(1,2),(3,2),(4,2),(5,2),(6,2)\}$
and
$B=\{(2,5),(5,2),(6,1),(1,6),(3,4),(4,3)\}$
$\therefore P(A)=\frac{11}{36}, P(B)=\frac{6}{36}$
$P(A \cap B)=\frac{2}{36}$
$\therefore $ Required probability
$P(A / B)=\frac{P(A \cap B)}{P(B)}=\frac{2 / 36}{6 / 36}=\frac{2}{6}=\frac{1}{3}$