Three dice are thrown simultaneously and the sum of the numbers appeared on them is noted. If $A$ is the event of getting a sum greater than $14$ and $B$ is the event of getting a sum which is a multiple of $3$,then $P(A \cap \overline{B}) + P(\overline{A} \cap B) = $

Four persons $A, B, C$ and $D$ throw an unbiased die,turn by turn in succession till one gets an even number and wins the game. What is the probability that $A$ wins the game if $A$ begins?

Consider the $6 \times 6$ square grid in the figure. Let $A_1, A_2, \ldots, A_{49}$ be the points of intersection (dots in the picture) in some order. We say that $A_i$ and $A_j$ are friends if they are adjacent along a row or along a column. Assume that each point $A_i$ has an equal chance of being chosen.
$(1)$ Let $p_i$ be the probability that a randomly chosen point has $i$ many friends,$i=0, 1, 2, 3, 4$. Let $X$ be a random variable such that for $i=0, 1, 2, 3, 4$,the probability $P(X=i)=p_i$. Then the value of $7 E(X)$ is
$(2)$ Two distinct points are chosen randomly out of the points $A_1, A_2, \ldots, A_{49}$. Let $p$ be the probability that they are friends. Then the value of $7 p$ is

If $n$ positive integers are taken at random and multiplied together,the probability that the last digit of the product is $2, 4, 6$ or $8$ is:

For any two events $A$ and $B$ in a sample space,which of the following is always true?

There are four machines and it is known that exactly two of them are faulty. They are tested one by one,in a random order,until both the faulty machines are identified. The probability that only two tests are needed is: