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.

If $P(A) = 0.65,\,\,P(B) = 0.15,$ then $P(\bar A) + P(\bar B) = $

On her vacations Veena visits four cities $(A,\,B ,\, C$ and $D$ ) in a random order. What is the probability that she visits $A$ before $B$ and $B$ before $C ?$

Describe the sample space for the indicated experiment: A coin is tossed and then a die is rolled only in case a head is shown on the coin.

Suppose that a die (with faces marked $1$ to $6$) is loaded in such a manner that for $K = 1, 2, 3…., 6$, the probability of the face marked $K$ turning up when 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

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 $C$ are mutually exclusive