The chance of getting a doublet with $2$ dice is

The probabilities of a problem being solved by two students are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. What is the probability that the problem is solved?

Three fair coins are tossed. If both heads and tails appear,then the probability that exactly one head appears is:

Two dice are thrown. What is the probability that the sum of the numbers appearing on the two dice is $11$,given that $5$ appears on the first die?

For independent events $A_1, A_2, \dots, A_n$, $P(A_i) = \frac{1}{i + 1}$ for $i = 1, 2, \dots, n$. Then the probability that none of the events will occur is:

One bag contains $4$ white balls and $2$ black balls; another contains $3$ white balls and $5$ black balls. If one ball is drawn from each bag,find the probability that both are white.