A problem in Mathematics is given to three students $A, B, C$ and their respective probability of solving the problem is $\frac{1}{2} , \frac{1}{3} $ and $\frac{1}{4}$. Probability that the problem is solved is

A bag contains $3$ red and $7$ black balls, two balls are taken out at random, without replacement. If the first ball taken out is red, then what is the probability that the second taken out ball is also red

A bag contains $4$ white, $5$ black and $6$ red balls. If a ball is drawn at random, then what is the probability that the drawn ball is either white or red

Let $E _{1}, E _{2}, E _{3}$ be three mutually exclusive events such that $P \left( E _{1}\right)=\frac{2+3 p }{6}, P \left( E _{2}\right)=\frac{2- p }{8}$ and $P \left( E _{3}\right)$ $=\frac{1- p }{2}$. If the maximum and minimum values of $p$ are $p _{1}$ and $p _{2}$, then $\left( p _{1}+ p _{2}\right)$ is equal to.

A determinant is chosen at random. The set of all determinants of order $2$ with elements $0$ or $1$ only. The probability that value of the determinant chosen is positive, is

Let $M$ be the maximum value of the product of two positive integers when their sum is $66$. Let the sample space $S=\left\{x \in Z: x(66-x) \geq \frac{5}{9} M\right\}$ and the event $A=\{ x \in S : x$ is a multiple of $3$ $\}$. Then $P ( A )$ is equal to