If $A$ and $B$ are two events of a random experiment such that $P(\bar{A})=\frac{2}{3}, P(B)=\frac{4}{15}$ and $P(A \cap \bar{B})=\frac{1}{5}$, then $\sqrt{195[P(B \mid(A \cup \bar{B}))+P(A \cup B)]}=$

A random variable $X$ has the range $\{0,1,2, \ldots$.$\} . If P(X=r)=k(1+r) 3^{-r}$, for $r=0,1,2, \ldots$ where $k>0$ is a real number, then $P(X=0)+P(X=1)+P(X=2)=$

In an experiment a person gets success $\alpha$ times out of $\beta$ trails. If the experiment consists of $n$ trials, then the probability that he fails at least $(n-1)$ times is

When two dice are thrown, the probability of getting a prime number on die and a composite number on the other is