$A$ biased die is marked with numbers $2, 4, 8, 16, 32, 32$ on its faces. The probability of getting a face with mark $n$ is $\frac{1}{n}$. If the die is thrown thrice,then the probability that the sum of the numbers obtained is $48$ is:

Box-$I$ contains $3$ cards bearing numbers $1, 2, 3$; Box-$II$ contains $5$ cards bearing numbers $1, 2, 3, 4, 5$ and Box-$III$ contains $7$ cards bearing numbers $1, 2, 3, 4, 5, 6, 7$. One card is drawn at random from each of the boxes. If $x_i$ is the number on the card drawn from the $i^{\text{th}}$ box,$i=1, 2, 3$,then the probability that $x_1+x_2+x_3$ is odd is equal to

Bag $A$ contains $3$ white and $4$ red balls,bag $B$ contains $4$ white and $5$ red balls,and bag $C$ contains $5$ white and $6$ red balls. If one ball is drawn at random from each of these three bags,then the probability of getting one white and two red balls is

Given two independent events,if the probability that exactly one of them occurs is $\frac{26}{49}$ and the probability that none of them occurs is $\frac{15}{49}$,then the probability of the more probable of the two events is (in $/7$)

If a man throws a die until he gets a number bigger than $3$,then the probability that he gets a $5$ in his last throw is

Let $A$ and $B$ be independent events such that $P(A)=p$ and $P(B)=2p$. The largest value of $p$, for which $P(\text{exactly one of } A, B \text{ occurs}) = \frac{5}{9}$, is: