Probability

Let $A$ and $B$ be two non-null events such that $A\subset\,B.$ Then, which of the following statements is always correct?

If$A$and$B$are two events such that $P\left(A\cup\,B\right)=P\left(A\cap\,B\right)$, then the incorrect statement amongst the following statements is :

For three events, $A,B$ and $C,$$P$(Exactly one of$A$or$B$occurs) $=P$(Exactly one of$B$or$C$occurs) $=P$(Exactly one of $C$or$A$occurs) $=\frac{1}{4}$ and $P$(All the three events occur simultaneously) $=\frac{1}{16}.$…

If$A$and$B$are two events such that$P\left(A\right)=\frac{1}{3},P\left(B\right)=\frac{1}{5}$and$P\left(A\cup\,B\right)=\frac{1}{2}$, then$P\left(AB^{'}\right)+P\left(BA^{'}\right)$is equal to

Let$S=\left{1,2,3,\ldots\,,2022\right}$. Then the probability, that a randomly chosen number$n$from the set$S$such that$HCF\left(n,2022\right)=1$, is

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and let $a=P(X=3),b=P(X\geq\,3)$ and $c=$ $P(X\geq\,6\mid\,X>3)$. Then $\frac{b+c}{a}$ is equal to

Let $A$ and $B$be two events such that $P\left(B\mid\,A\right)=\frac{2}{5}$, $P\left(A\mid\,B\right)=\frac{1}{7}$ and $P\left(A\cap\,B\right)=\frac{1}{9}$. Consider…

In a bombing attack, there is $50%$ chance that a bomb will hit the target. At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure…

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac{1}{3}$ and $\frac{2}{3}$ respectively. Let $x$ be the number of matches that the team wins, and $y$ be the number of…

Let the probability of getting head for a biased coin be$\frac{1}{4}$. It is tossed repeatedly until a head appears. Let$N$be the number of tosses required. If the probability that the equation$64x^{2}+5Nx+1=0$has no…

From the first 100 natural numbers, two numbers first $a$ and then $b$ are selected randomly without replacement. If the probability that $a-b \geqslant 10$ is $\frac{m}{n}, \operatorname{gcd}(m, n)=1$, then $m+n$ is…

A coin is tossed $8$ times. If the probability that exactly $4$ heads appear in the first six tosses and exactly $3$ heads appear in the last five tosses is $p$, then $96p$ is equal to _____?