$A$ 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 computer program generate only the digits $0$ and $1$ to form a string of binary numbers. The probability of occurrence of $0$ at even places is $\frac{1}{2}$ and the probability of occurrence of $0$ at odd places is $\frac{1}{3}$. Then the probability that $'10'$ is followed by $'01'$ is equal to:

If four positive integers are selected randomly from the set of positive integers,then the probability that the unit digit of their product is $1, 3, 7,$ or $9$ is:

Two dice are thrown and two coins are tossed simultaneously. The probability of getting prime numbers on both the dice along with a head and a tail on the two coins is

Let $a, b, c \in \{1, 2, 3, 4\}$. If the probability that $ax^2 + 2\sqrt{2}bx + c > 0$ for all $x \in R$ is $m/n$, where $gcd(m, n) = 1$, then $m + n$ is equal to . . . . . . .

Two balls are drawn at random with replacement from a box containing $10$ black and $8$ red balls. Find the probability that one of them is black and the other is red.