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)]} = $

An ordinary cube has four blank faces,one face marked $2$,and another marked $3$. The probability of obtaining a total of exactly $12$ in $5$ throws 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:

$12$ balls are distributed among $3$ boxes. The probability that the first box will contain exactly $3$ balls is:

Cards are drawn one by one without replacement from a pack of $52$ cards. The probability that $10$ cards will precede the first ace is

If $S$ is the sample space of a random experiment $\xi$ and $P$ is a probability function defined on the power set $\mathcal{P}(S)$ of $S$,then which one of the following is not satisfied by $P$?
$(i)$ $P(\phi) = 0$
(ii) If $E^c$ is the complementary event of $E$,then $P(E^c) = 1 - P(E)$
(iii) $0 \leq P(E) \leq 1, \forall E \subseteq S$
(iv) If $E_1 \subseteq E_2$,then $P(E_2) \leq P(E_1)$