$A$ single letter is selected at random from the word $PROBABILITY$. The probability that the selected letter is a vowel is

If $P(A) = 2/3$,$P(B) = 1/2$ and $P(A \cup B) = 5/6$,then events $A$ and $B$ are

$5$ persons $A, B, C, D$ and $E$ are in a queue at a shop. The probability that $A$ and $E$ are always together is:

Three persons work independently on a problem. If the respective probabilities that they will solve it are $\frac{1}{3}$,$\frac{1}{4}$,and $\frac{1}{5}$,then the probability that none can solve it is:

Three $6$-faced dice are thrown together. The probability that exactly two dice show the same number on them is:

One bag contains $5$ white and $3$ black balls and a second bag contains $2$ white and $4$ black balls. One ball is drawn from the first bag and placed unseen in the second bag. What is the probability that a ball now drawn from the second bag is black?