If $X_1, X_2, \ldots, X_n$ are $n$ independent events such that $P(X_r) = \frac{1}{r+1}$ for $r = 1, 2, \ldots, n$,then the probability that none of the $n$ events occur is

The probability that a number selected at random from the set of numbers $(1, 2, 3, \ldots, 100)$ is a cube,is

$A$ card is drawn at random from a pack of $52$ cards. What is the probability that the drawn card is neither a heart nor a king?

Three coins are tossed. Describe three events which are mutually exclusive but not exhaustive.

Three persons work independently on a problem. If the respective probabilities that they will solve it are $\frac{1}{3}$,$\frac{1}{4}$,and $\frac{1}{5}$,then the probability that none of them can solve it is:

Two dice are thrown. The events $A$,$B$,and $C$ are as follows:
$A$: getting an even number on the first die.
$B$: getting an odd number on the first die.
$C$: getting the sum of the numbers on the dice $\leq 5$.
State true or false: (give reason for your answer)
Statement: $A$ and $B$ are mutually exclusive.