There are $100$ lottery tickets numbered from $1$ to $100$. If a ticket is drawn at random,find the probability that the number on it is a multiple of $3$ or $5$.

If two events $E_1$ and $E_2$ are such that $P(E_1 \cup E_2) = \frac{5}{8}$, $P(\bar{E}_1) = \frac{3}{4}$, and $P(E_2) = \frac{1}{2}$, then $E_1$ and $E_2$ are:

$A$ and $B$ are events such that $P(A)=0.42$,$P(B)=0.48$ and $P(A \cap B)=0.16$. Determine $P(A \cup B)$.

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 $P(\text{not } E \text{ and not } F)$.

Three critics review a book. For the three critics,the odds in favour of the book are $(5: 2)$,$(4: 3)$ and $(3: 4)$ respectively. The probability that the majority is in favour of the book is

The probability of an event happening is the square of the probability of a second event,but the odds against the first are the cube of the odds against the second. The probabilities of the events are: