Find $P(E | F)$ for a coin tossed three times,where $E: \text{head on third toss}$,$F: \text{heads on first two tosses}$.

Prove that if $E$ and $F$ are independent events,then so are the events $E$ and $F^{\prime}$.

If $80 \%$ of flights depart on time, $70 \%$ of flights arrive on time and $65 \%$ of flights depart on time and arrive on time, then the probability that a flight that has just departed on time will arrive on time is

If $A$ and $B$ are two independent events,then $P\left( \frac{A}{B} \right) = $

$A$ and $B$ are two events such that $P(A) = 0.8$,$P(B) = 0.6$,and $P(A \cap B) = 0.5$. Then the value of $P(A/B)$ is:

$A$ 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: