Finite Sample Space

With the help of a Venn diagram determine whether the following events $E,$ and $F$ of a sample space $S=\left\{1,2,3,4\right\}$ are mutually exclusive events or mutually inclusive events.

$E=\left\{1,2\right\},$ and $F=\left\{3\right\}$

With the help of a Venn diagram determine whether the following events $E,$ and $F$ of a sample space $S=\left\{1,2,3,4\right\}$ are mutually exclusive events or mutually inclusive events.

$E=\left\{1,2\right\},$ and $F=\left\{4\right\}$

With the help of a Venn diagram determine whether the following events $E,$ and $F$ of a sample space $S=\left\{1,2,3,4\right\}$ are mutually exclusive events or mutually inclusive events.

$E=\left\{1,2\right\},$ and $F=\left\{2,4\right\}$

Which one of the following represents males, boys, foot-ball players?

If $A$ and $B$ are mutually exclusive events $P\left(A\right)=\frac{3}{8}$ and $P\left(B\right)=\frac{1}{8}$, then find $P\left(\overline{A}\right)$.

Given two independent events $A$ and $B$ such that $P\left(A\right)=0.3, P\left(B\right)=0.6$. Find $P\left(A \mathrm{or} B\right)$.

For next three question please follow the same

$A$ is a set containing $10$ elements. A subset $P$ of $A$ is chosen at random and the set $A$ is reconstructed by replacing the elements of $P$.Another subset $Q$ of $A$ is now chosen at random. Then the probability that if :

$\text{P}\cup \text{Q}=\text{A}$ is

The probability of a person remains alive for the next $25$ years is $4/5$ and the probability that his wife remain alive for the same $25$ years is $3/4$, Find the probabilities that only wife remain alive for the next $25$ years.

An urn contains $5$ white, $7$ red and $8$ black balls. If four balls are drawn with replacement then what is the probability that at least three balls are white.

An urn contains $5$ white, $7$ red and $8$ black balls. If four balls are drawn with replacement then what is the probability that none of the balls are white.

An urn contains $5$ white, $7$ red and $8$ black balls. If four balls are drawn with replacement then what is the probability that only three balls are white.

Probability of solving specific problem independently by $\mathrm{A}$ and $\mathrm{B}$ are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that exactly one of them solves the problem.

Two coins are tossed. Find the probability of getting two heads when it is known that one head has already occurred.

If $\mathrm{A}$ and $\mathrm{B}$ are independent events such that $P\left(A\right)=0.3$ and $P\left(B\right)=0.4$, then find $P\left(B|A\right)$.

If $\mathrm{A}$ and $\mathrm{B}$ are independent events and $P\left(A\right)=0.3, P\left(B\right)=0.6$ then find $P(A\cap B)$.

If $\mathrm{A}$ and $\mathrm{B}$ are independent events such that $P\left(A\right)=0.3$ and $P\left(B\right)=0.4$ then find $P(A\cup B)$.

A die is thrown and a card is selected at random from a deck of $52$ playing cards. The probability of getting an even number on the die and a spade card is

A speaks the truth in $60\%$ cases and B in $80\%$ cases. The probability that they will contradict each other in describing a single event is

If $A, B \& C$  are mutually exclusive and exhaustive events of a random experiment such that $P\left(B\right)=\frac{3}{2}P\left(A\right)$ and $P\left(C\right)=\frac{1}{2}P\left(B\right),$ then $P\left(A\cup C\right)$ equals to

Let F denote the set of all onto functions from $A=\left\{a_1 , a_2 , a_3 , a_4\right\} to B=\left\{x,y,z\right\}.$ A function f is chosen at random from F. The probability that $f^{-1}\left(x\right)$ consists of exactly two elements is