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

Let $N$ be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N - 2, \sqrt{3N}, N + 2$ are in geometric progression be $\frac{k}{48}$. Then the value of $k$ is

$A$ box $P$ contains one white ball,three red balls and two black balls. Another box $Q$ contains two white balls,three red balls and four black balls. If one ball is drawn at random from each one of the two boxes,then the probability that the balls drawn are of different colour 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 whether the following statement is true or false and provide a reason:
Statement: $A = B^{\prime}$

When three fair dice are tossed simultaneously,find the probability that the integers on all three dice are the same.

Let the sum of two positive integers be $24$. If the probability,that their product is not less than $\frac{3}{4}$ times their greatest positive product,is $\frac{m}{n}$,where $\operatorname{gcd}(m, n)=1$,then $n-m$ equals :