Consider the experiment of throwing a die. If a multiple of $3$ comes up,throw the die again,and if any other number comes,toss a coin. Find the conditional probability of the event 'the coin shows a tail',given that 'at least one die shows a $3$'.

$A$ black and a red dice are rolled. Find the conditional probability of obtaining the sum $8$,given that the red die resulted in a number less than $4$.

For a biased die,the probabilities for different faces to turn up are

$Face$	$1$	$2$	$3$	$4$	$5$	$6$
$P(F)$	$0.2$	$0.22$	$0.11$	$0.25$	$0.05$	$0.17$

The die is tossed and you are told that either face $4$ or face $5$ has turned up. The probability that it is face $4$ is

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 $3$?

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 $6 P(A) = 8 P(B) = 14 P(A \cap B) = 1$,then $P(A' \mid B) = $ . . . . . . .