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

Two dice are thrown. The events $A, B$ and $C$ are as follows:
$A:$ getting an even number on the first die.
$B:$ getting an odd number on the first die.
$C:$ getting the sum of the numbers on the dice $\leq 5$.
State true or false: (give reason for your answer)
Statement: $A$ and $B$ are mutually exclusive and exhaustive.

$A$ container holds two balls. Both balls are black. If one white ball is added to the container and then one ball is drawn at random from the container,what is the probability that it is white?

In each of the following experiments,specify the appropriate sample space: $A$ boy has a $1$ rupee coin,a $2$ rupee coin,and a $5$ rupee coin in his pocket. He takes out two coins out of his pocket,one after the other.

If two dice are thrown together,then the probability that the sum of the numbers appearing on them is a prime number is:

There are two children in a family. The probability that both of them are boys is