Let $E$ and $F$ be two independent events. The probability that both $E$ and $F$ happen is $\frac{1}{12}$ and the probability that neither $E$ nor $F$ happens is $\frac{1}{2}$. Then a value of $\frac{P(E)}{P(F)}$ is

Two persons $A$ and $B$ take part in a shooting competition. $A$ can hit the target with a probability of $0.6$. $B$ can hit the target with a probability of $0.8$. $A$ takes the first shot,after which they shoot alternately. The probability that $A$ wins the competition is

If $A$ and $B$ are independent events of a random experiment such that $P(A \cap B) = \frac{1}{6}$ and $P(\bar{A} \cap \bar{B}) = \frac{1}{3}$,then $P(A)$ is equal to (Here,$\bar{E}$ is the complement of the event $E$)

There are four machines and it is known that exactly two of them are faulty. They are tested one by one,in a random order,until both the faulty machines are identified. The probability that only two tests are needed is:

Let $A, B$ and $C$ be three events such that the probability that exactly one of $A$ and $B$ occurs is $(1-k)$,the probability that exactly one of $B$ and $C$ occurs is $(1-2k)$,the probability that exactly one of $C$ and $A$ occurs is $(1-k)$ and the probability that all $A, B$ and $C$ occur simultaneously is $k^2$,where $0 < k < 1$. Then the probability that at least one of $A, B$ and $C$ occurs is:

There are four fair dice $D_1, D_2, D_3$ and $D_4$. Each has six faces numbered $1, 2, 3, 4, 5$ and $6$. They are rolled one by one. What is the probability that the number shown on $D_4$ is equal to at least one of the numbers shown on $D_1, D_2$ and $D_3$ (in $/216$)?