If $E$ and $F$ are events such that $P(E) = \frac{1}{4}$,$P(F) = \frac{1}{2}$ and $P(E \cap F) = \frac{1}{8}$,find $P(E \cup F)$.

$A$ card is drawn at random from a well-shuffled deck of $52$ cards. What is the probability that the card is red or a queen?

$A$ person $P$ speaks the truth in $75\%$ of cases and another person $R$ in $80\%$ of cases. What is the probability that they are likely to contradict each other in narrating the same event?

In class $XI$ of a school,$40\%$ of the students study Mathematics and $30\%$ study Biology. $10\%$ of the class study both Mathematics and Biology. If a student is selected at random from the class,find the probability that they will be studying Mathematics or Biology.

The probabilities that $A$ and $B$ speak the truth are $\frac{4}{5}$ and $\frac{3}{4}$ respectively. The probability that they contradict each other when asked to speak on a fact is

The odds against solving a problem by $A$ and $B$ are $3:2$ and $2:1$ respectively. The probability that the problem will be solved is: