The probability that a randomly chosen $2 \times 2$ matrix with all the entries from the set of first $10$ primes is singular,is equal to

If $A$ and $B$ are independent events of a random experiment such that $P(A \cap B)=\frac{1}{6}$ and $P(\bar{A} \cap \bar{B})=\frac{1}{3}$, then $P(A)$ is equal to (Here, $\bar{E}$ is the complement of the event $E$)

In a random experiment,a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to

$A$ fair die is thrown until $2$ appears. Then the probability that $2$ appears in an even number of throws is

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

$A$ pack of cards contains $4$ aces,$4$ kings,$4$ queens,and $4$ jacks. Two cards are drawn at random. The probability that at least one of these is an ace is: