Independent Events

Let  $E^c$  denote the complement of an event $E$. Let $E,F,G$ be pairwise independent events with  $P\left(G\right)>0$ and $P\left(E\cap F\cap G\right)=0$. Then  $P\left(E^c\cap F^c|G\right)$ equals 

Two fair dice are rolled. Let $X$ be the event that the first die shows an even number and $Y$ be the event that the second die shows an odd number. The two events $X$ and $Y$ are:

$2$ unbiased die are thrown independently. $A$ is the event such that the number on the first die is greater than the number on the second die. $B$ is the event such that the number on the first die is even and number on the second die is odd. $C$ is the event such that first die shows odd number and second die shows even number, then

Events $E$ and $F$ are independent. Find $P\left(F\right)$, if $P\left(E\right)=\frac{2}{5}$ and $P\left(E\cup F\right)=\frac{3}{5}.$

If $A$ and $B$ are two independent events such that $P\left(\overline{A}\right)=0.75,P\left(A\cup B\right)=0.65$, and $P\left(B\right)=x$, then find the value of $x$:

Given that the events $A$ and $B$ are such that $P\left(A\right)=\frac{1}{2}$, $P(A\cup B)=\frac{3}{5}$ and $P\left(B\right)=p$. Find $p$ if the events are independent.

Out of a pack of ten cards numbered $1$to $10,$ a boy draws a card at random and keeps it back. Then a girl draws a card at random from the same pack. If the boy's card reads $m,$ and the girl's card reads $n,$ then what is the probability that $m>n$, given that $m$ is even$?$

On rolling a dice $6$ times probability of event of obtaining even number and event of obtaining prime numbers are mutually independent or dependent?

If $A$ and $B$ are two independent events, then the probability of occurrence of at least one of $A$ and $B$ is given by $1-P\left(A^{\text{'}}\right)P\left(B^{\text{'}}\right)$

Prove that if $E$ and $F$ are independent events, then so are the events $E$ and $F^{\text{'}}$.

An unbiased die is thrown twice. Let the event $A$ be 'odd number on the first throw' and $B$ the event 'odd number on the second throw'. Check the independence of the events $A$ and $B$.

A die is thrown. If $E$ is the event 'the number appearing is a multiple of $3\text{'}$ and $F$ be the event 'the number appearing is even' then find whether $E$ and $F$ are independent?

If $A$ and $B$ are independent events such that $P\left(A\right)=p, P\left(B\right)=2p$ and $P$(Exactly one of $A$ and $B$)$=\frac{5}{9}$, then $p=$

For two events $E$ and $F$, given that $P\left(E\right)=\frac{1}{3},P\left(F\right)=q$ and $P\left(E\cup F\right)=\frac{3}{7}$. If $E$ and $F$ are independent events, then $q$ is equal to

If $E$ and $F$ are independent events such that $P\left(E\right)=\frac{1}{3}$ and $P\left(F\right)=\frac{1}{6}$, then the probability that neither $E$ nor $F$ occurs is

A six-faced unbiased die is thrown until a number greater than $4$ appears. The probability that this occurs on the $n$-th throw, where $n$ is an even integer, is

An event $A$ is independent of itself if and only if $P\left(A\right)$ is

If $A$ and $B$ be independent events with $P\left(A\right)=\frac{1}{3}$ and $P\left(B\right)=\frac{2}{7}$, then the value of $P\left(A/B^C\right)$ is

If $A$ and $B$ are two independent events such that $P\left(A\right)=\frac{3}{10}$ and $P\left(A\cup B\right)=\frac{4}{5},$ then $P\left(A\cap B\right)$ is equal to

Two fair dice are rolled. Let $X$ be the event that the first die shows an even number and $Y$ be the event that the second die shows an odd number. The two events $X$ and $Y$ are: