An urn contains $5$ red and $5$ black balls. $A$ ball is drawn at random,its colour is noted and is returned to the urn. Moreover,$2$ additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?

If $P(E)=0.8, P(F)=0.5$ and $P(F \mid E)=0.4$ then,$P(E \mid F)=$ . . . . . . .

Find $P(E | F)$ for a die thrown three times,where $E: 4$ appears on the third toss,and $F: 6$ and $5$ appear respectively on the first two tosses.

$A$ candidate takes three tests in succession and the probability of passing the first test is $p$. The probability of passing each succeeding test is $p$ if he passes the preceding one, and $\frac{p}{2}$ if he fails the preceding one. The candidate is selected if he passes at least two tests. The probability that the candidate is selected is:

If $C$ and $D$ are two events such that $C \subset D$ and $P(D) \neq 0$,then which of the following is true?

If $A$ and $B$ are any two events such that $P(A) + P(B) - P(A \cap B) = P(A)$, then $\dots \dots \dots$