Four persons independently solve a certain problem correctly with probabilities $\frac{1}{2}, \frac{3}{4}, \frac{1}{4}, \frac{1}{8}$. Then the probability that the problem is solved correctly by at least one of them is

$A$ coin is tossed. If it shows a tail,we draw a ball from a box which contains $2$ red and $3$ black balls. If it shows a head,we throw a die. Find the sample space for this experiment.

Suppose that a die (with faces marked $1$ to $6$) is loaded in such a manner that for $K = 1, 2, 3, 4, 5, 6$,the probability of the face marked $K$ turning up when the die is tossed is proportional to $K$. The probability of the event that the outcome of a toss of the die will be an even number is equal to

An integer $x$ is chosen from $1$ to $50.$ Then the probability that $x + \frac{336}{x} \le 50$ is

$A$ batch of pencils contains $12$ good,$6$ slightly defective,and $2$ very defective pencils. If a pencil is selected at random,what is the probability that it is a non-defective pencil?

Three coins are tossed. Describe two events which are mutually exclusive.