If $P(A) = \frac{6}{11}$,$P(B) = \frac{5}{11}$ and $P(A \cup B) = \frac{7}{11}$,find $P(A | B)$.

An urn $A$ contains $3$ white and $5$ black balls. Another urn $B$ contains $6$ white and $8$ black balls. $A$ ball is picked from $A$ at random and then transferred to $B$. Then, a ball is picked at random from $B$. The probability that it is a white ball is

Two dice are thrown and the sum of the numbers appearing on the dice is observed to be a multiple of $4$. If $p$ is the conditional probability that number $4$ has appeared at least once, then $3p + 2 =$

An electronic assembly consists of two subsystems,$A$ and $B$. From previous testing procedures,the following probabilities are known:
$P(A \text{ fails}) = 0.2$
$P(B \text{ fails alone}) = 0.15$
$P(A \text{ and } B \text{ fail}) = 0.15$
Evaluate the probability $P(A \text{ fails } | \text{ } B \text{ has failed})$.

$A$ cubical die with faces marked $1, 2, 3, ..., 6$ is tossed such that the probability of throwing the number $t$ is proportional to $t^2$. The probability that the number $5$ has appeared,given that the number turned up is not even,is:

Two dice are thrown. If it is known that the sum of numbers on the dice was less than $6$,the probability of getting a sum as $3$ is: