$A , B, C$ try to hit a target simultaneously but independently. Their respective probabilities of hitting targets are $\frac{3}{4},\frac{1}{2},\frac{5}{8}$. The probability that the target is hit by $A$ or $B$ but not by $C$ is

If $A$ and $B$ are two events such that $P(A) = 0.4$ , $P\,(A + B) = 0.7$ and $P\,(AB) = 0.2,$ then $P\,(B) = $

Two dice are thrown. What is the probability that the sum of the numbers appearing on the two dice is $11$, if $5$ appears on the first

A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing $15$ oranges out of which $12$ are good and $3$ are bad ones will be approved for sale.

Consider three sets $E_1=\{1,2,3\}, F_1=\{1,3,4\}$ and $G_1=\{2,3,4,5\}$. Two elements are chosen at random, without replacement, from the set $E _1$, and let $S _1$ denote the set of these chosen elements.

Let $E_2=E_1-S_1$ and $F_2=F_1 \cup S_1$. Now two elements are chosen at random, without replacement, from the set $F_2$ and let $S_2$ denote the set of these chosen elements.

Let $G _2= G _1 \cup S _2$. Finally, two elements are chosen at random, without replacement, from the set $G _2$ and let $S _3$ denote the set of these chosen elements.

Let $E_3=E_2 \cup S_3$. Given that $E_1=E_3$, let $p$ be the conditional probability of the event $S_1=\{1,2\}$. Then the value of $p$ is

The odds against a certain event is $5 : 2$ and the odds in favour of another event is $6 : 5$. If both the events are independent, then the probability that at least one of the events will happen is