$A$ and $B$ toss a fair coin each simultaneously $50$ times. The probability that both of them will not get tail at the same toss is

There are $2$ shelves. One shelf has $5$ Physics and $3$ Biology books,and the other has $4$ Physics and $2$ Biology books. What is the probability of drawing a Physics book?

In a game,two players $A$ and $B$ take turns throwing a pair of fair dice,starting with player $A$. The total score on the two dice in each throw is noted. $A$ wins the game if he throws a total of $6$ before $B$ throws a total of $7$,and $B$ wins the game if he throws a total of $7$ before $A$ throws a total of $6$. The game stops as soon as either of the players wins. The probability of $A$ winning the game is:

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 and $P(A) = p, P(B) = 2p$,and $P(\text{exactly one from } A \text{ and } B) = \frac{5}{9}$,then find the value of $p$.

$A$ bag contains $4$ red and $5$ black balls. Another bag contains $3$ red and $6$ black balls. If one ball is drawn from the first bag and two balls are drawn from the second bag at random,the probability that out of the three balls,two are black and one is red,is