$A$ fair die is rolled. Consider events $E=\{1,3,5\}, F=\{2,3\},$ and $G=\{2,3,4,5\}$. Find $P(E | G)$ and $P(G | E)$.

Find $P(E | F)$ if a mother,father,and son line up at random for a family picture,where $E$ is the event that the son is on one end and $F$ is the event that the father is in the middle.

When $2$ dice are thrown, it is observed that the sum of the numbers appearing on the top faces of both dice is a prime number. What is the probability that at least one of the numbers in the pair is a multiple of $3$?

If $A$ and $B$ are two events of a sample space $S$ such that $P(A)=0.2$,$P(B)=0.6$ and $P(A \mid B)=0.5$,then $P(A^{\prime} \mid B) = $

Out of $50$ tickets numbered $00, 01, 02, \dots, 49$,one ticket is drawn at random. Let $A$ be the event that the sum of the digits on the ticket is $8$,and $B$ be the event that the product of the digits is $0$. Find the conditional probability $P(A|B)$.

Consider three sets $E_1=\{1,2,3\}, F_1=\{1,3,4\}$ and $G_1=\{2,3,4,5\}$. Two elements are chosen at random,without replacement,from the set $E_1$,and let $S_1$ denote the set of these chosen elements.
Let $E_2=E_1-S_1$ and $F_2=F_1 \cup S_1$. Now two elements are chosen at random,without replacement,from the set $F_2$ and let $S_2$ denote the set of these chosen elements.
Let $G_2=G_1 \cup S_2$. Finally,two elements are chosen at random,without replacement,from the set $G_2$ and let $S_3$ denote the set of these chosen elements.
Let $E_3=E_2 \cup S_3$. Given that $E_1=E_3$,let $p$ be the conditional probability of the event $S_1=\{1,2\}$. Then the value of $p$ is