Consider the following for the two (02) items that follow:
The probabilities that A, B and C become managers are 3/10 1/2 and 4/5 respectively. The probabilities that bonus scheme will be introduced if A, B and C become managers are 4/9, 2/9 and 1/3 respectively.

If the bonus scheme has been introduced, then what is the probability that the manager appointed was B?

If $P(A)=1/3$, $P(B)=1/2$ and $P(A\cap B)=1/4$, then what is the value of $P(A|B{}^C)$?

If $P(A)=1/3$, $P(B)=1/2$ and $P(A\cap B)=1/4$, then what is the value of $P(A{}^C\cap B{}^C)$?

If two fair dice are tossed, then what is the probability that the sum of the numbers on the faces of the dice is strictly greater than 7?