Probability

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) $=\frac{1}{16}.$…

Let $A$ be the event that the absolute difference between two randomly chosen real numbers in the sample space $\left[0,60\right]$ is less than or equal to $a$. If $P\left(A\right)=\frac{11}{36}$, then a is equal to…

All five letter words are made using all the letters $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}$ and arranged as in an English dictionary with serial numbers. Let the word at serial number $n$ be…

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

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

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar{X}$ and $\bar{Y}$ are the means of…

The urns$A,B$and$C$contains$4$red,$6$black;$5$red,$5$black and$\lambda\,$red,$4$black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and theprobability that it is…

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 $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked $1,2,3,4$. If the probability that $a x^2+b x+c=0$ has all real roots is…

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 :

$25%$ of the population are smokers. A smoker has $27$ times more chances to develop lung cancer then a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is…

Let$A$and$E$be any two events with positive probabilities Statement I: $P\left(E/A\right)\geq\,P\left(A/E\right)P\left(E\right).$ Statement II: $P\left(A/E\right)\geq\,P\left(A\cap\,E\right).$