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

Assume that each born child is equally likely to be a boy or a girl. If a family has two children,what is the conditional probability that both are girls given that the youngest is a girl?

$A$ family has $3$ children. The probability that all the three children are girls,given that at least one of them is a girl is:

In a group of $14$ males and $6$ females, $8$ males and $3$ females are aged above $40 \text{ yr}$. The probability that a person selected at random from the group is aged above $40 \text{ yr}$, given that the selected person is a female, is:

Let $E^c$ denote the complement of an event $E$. Let $E, F, G$ be pairwise independent events with $P(G)>0$ and $P(E \cap F \cap G) = 0$. Then $P(E^c \cap F^c \mid G)$ equals

Let $B(\alpha, \beta, \gamma)$ represent that a bag $B$ contains $\alpha$ red balls, $\beta$ green balls, and $\gamma$ blue balls. Given $B_1(2, 3, 2)$, $B_2(3, 2, 2)$, $B_3(2, 2, 3)$. $A$ die is rolled. If the die shows $2, 3,$ or $5$, a ball is drawn from bag $B_1$. If the die shows $4$ or $6$, a ball is drawn from bag $B_2$. If the die shows $1$, a ball is drawn from bag $B_3$. Find the probability of drawing a green ball.