The probability that in a year of the $22^{nd}$ century chosen at random there will be $53$ Sundays is

For independent events ${A_1},\,{A_2},\,..........,{A_n},$ $P({A_i}) = \frac{1}{{i + 1}},$ $i = 1,\,\,2,\,......,\,\,n.$ Then the probability that none of the event will occur, is

A fair coin is tossed repeatedly. If tail appears on first four tosses then the probability of head appearing on fifth toss equals

Two dice are thrown and the sum of the numbers which come up on the dice is noted. Let us consider the following events associated with this experiment

$A:$ $^{\prime}$ the sum is even $^{\prime}$.
$B:$ $^{\prime}$the sum is a multiple of $3$$^{\prime}$
$C:$ $^{\prime}$the sum is less than $4 $$^{\prime}$
$D:$ $^{\prime}$the sum is greater than $11$$^{\prime}$.

Which pairs of these events are mutually exclusive ?

If three students $A, B, C$ independently solve a problem with probabilitities $\frac{1}{3},\frac{1}{4}$ and $\frac{1}{5}$ respectively, then the probability that the problem will be solved is

Let two fair dices $A$ and $B$ are thrown. Then the probability that number appears on dice $A$ is greater than number appears on dice $B$ is