Cards are drawn one by one without replacement from a pack of $52$ cards. The probability that $10$ cards will precede the first ace is

Let $A$ be the set of all $4$-digit natural numbers whose exactly one digit is $7$. Then the probability that a randomly chosen element of $A$ leaves a remainder of $2$ when divided by $5$ is ..... .

There are three bags $B_1$,$B_2$,and $B_3$ containing $2$ Red and $3$ White,$5$ Red and $5$ White,and $3$ Red and $2$ White balls respectively. $A$ ball is drawn from bag $B_1$ and placed in bag $B_2$,then a ball is drawn from bag $B_2$ and placed in bag $B_3$,then a ball is drawn from bag $B_3$. The number of ways in which this process can be completed,if the same colour balls are used in the first and second transfers (assume all balls to be distinct),is

Five different books are to be distributed among four students randomly. The probability that each child gets at least one book 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?

$P$ speaks truth in $70\%$ of the cases and $Q$ in $80\%$ of the cases. In what percent of cases are they likely to agree in stating the same fact (in $\%$)?