A die is thrown twice and the sum of the dots on the two faces is observed to be 7. The conditional probability that the number 2 has appeared at least once is:

Let $S$ be the sample space of pairs with sum 7:$S=\{(1,6),(6,1),(2,5),(5,2),(3,4),(4,3)\}$ Total number of outcomes $n(S)=6$.Let $E$ be the event that number 2 appears at least once.Favorable outcomes in $S$ are:$E=\{(2,5),(5,2)\}$ Total number of favorable outcomes $n(E)=2$.Conditional Probability $P(E|Sum=7)=\frac{n(E)}{n(S)}=\frac{2}{6}=\frac{1}{3}$.