Four fair dice $D_1, D_2, D_3$ and $D_4$,each having six faces numbered $1, 2, 3, 4, 5$ and $6$,are rolled simultaneously. The probability that $D_4$ shows a number appearing on at least one of $D_1, D_2$ and $D_3$ is

Two events $A$ and $B$ are given. If $P(A) = \frac{1}{4}$,$P(A|B) = \frac{1}{2}$,and $P(B|A) = \frac{2}{3}$,then what is the value of $P(B)$?

Let $A$ and $B$ be two events with $P(A) = \frac{1}{7}$,$P(A|B) = \frac{2}{3}$,and $P(B) = \frac{2}{7}$. Then,the value of $P(B|A)$ is

$A$ couple has two children. Find the probability that both children are males,if it is known that at least one of the children is male.

$A$ candidate takes three tests in succession and the probability of passing the first test is $p$. The probability of passing each succeeding test is $p$ if he passes the preceding one, and $\frac{p}{2}$ if he fails the preceding one. The candidate is selected if he passes at least two tests. The probability that the candidate is selected is:

Find $P(E | F)$ for a die thrown three times,where $E: 4$ appears on the third toss,and $F: 6$ and $5$ appear respectively on the first two tosses.