$A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of $5$ before $B$ throws a sum of $8$,and $B$ wins if he throws a sum of $8$ before $A$ throws a sum of $5$. The probability that $A$ wins,if $A$ makes the first 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:

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

One die has two faces marked $1$,two faces marked $2$,one face marked $3$ and one face marked $4$. Another die has one face marked $1$,two faces marked $2$,two faces marked $3$ and one face marked $4$. The probability of getting the sum of numbers to be $4$ or $5$,when both the dice are thrown together,is

$A$ computer program has two modules $X$ and $Y$ and errors in them occur independently. $X$ has an error with probability $0.1$ and $Y$ has an error with probability $0.3$. If an error in $X$ alone causes the program to crash with probability $0.5$, an error in $Y$ alone causes the program to crash with probability $0.7$, and an error in both $X$ and $Y$ causes the program to crash with probability $0.8$, then the probability that the program crashes is

$A$ man throws a fair coin repeatedly. He gets $10$ points for each head he throws and $5$ points for each tail he throws. If the probability that he gets exactly $30$ points is $\frac{m}{n}$, where $\text{gcd}(m, n) = 1$, then $m + n$ is equal to: