If $P(B) = \frac{3}{4}$, $P(A \cap B \cap \bar C) = \frac{1}{3}{\rm{ }}$ and $P(\bar A \cap B \cap \bar C) = \frac{1}{3},$ then $P(B \cap C)$ is

Let $S$ be a set containing n elements and we select $2$ subsets $A$ and $B$ of $S$ at random then the probability that $A \cup B = S$ and $A \cap B = \phi $ is

An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered $2, 3, 4,.......,12$ is picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$, is

If ${A_1},\,{A_2},...{A_n}$ are any $n$ events, then

If the odds in favour of an event be $3 : 5$, then the probability of non-occurrence of the event is

Check whether the following probabilities $P(A)$ and $P(B)$ are consistently defined $P ( A )=0.5$,  $ P ( B )=0.4$,  $P ( A \cap B )=0.8$