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.

Evaluate $P(A \cup B)$,if $2 P(A) = P(B) = \frac{5}{13}$ and $P(A|B) = \frac{2}{5}$.

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.

If $P(A) = \frac{7}{13}$,$P(B) = \frac{9}{13}$ and $P(A \cap B) = \frac{4}{13}$,evaluate $P(A | B)$.

$A$ 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 $P((E \cap F) | G)$.

$A, B$ are the events in a random experiment. If $P(A)=\frac{1}{2}, P(B)=\frac{1}{3}, P(A \cap B)=\frac{1}{4}$,then $P\left(\frac{A^{c}}{B^{c}}\right)+P\left(\frac{A}{B}\right)=$