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

A box contains four blue balls and three green balls. Judith and Gilles play a game with each taking it in turn to take a ball from the box, without replacement. The first player to take a green ball is the winner. Judith plays first. Find the probability that she wins. The game is now changed so that the ball chosen is replaced after each turn. Judith still plays first. Determine whether the probability of Judith winning has changed.