Conditional Probability and Multiplication Theorem

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:

If $F$ and $E$ are events with $P\left(E\right)=\frac{3}{5},P\left(F\right)=\frac{3}{10}$ and $P(E\cap F)=\frac{1}{5}$, then the events $E$ and $F$ are independent.

A coin is tossed and a die is rolled. The probability that the coin shows head and the die shows $3$ is

If $A$ and $B$ are independent events such that $P\left(A\right)=0.3$ and $P\left(B\right)=0.4$ and $P(A\cup B)=\frac{k}{100}$, then the value of $k$ is

If $P\left(A\right)=\frac{7}{13},P\left(B\right)=\frac{9}{13}\text{ and }P(A\cap B)=\frac{4}{13},$ find $P(A/B)$.

If $A$ and $B$ are two independent events with $P\left(A\right)=0.2$ and $P\left(B\right)=0.5$, then the value of $P(A\cup B)$ is

A problem in calculus is given to two students $A$ and $B$, whose chances of solving it are $\frac{1}{3}$ and $\frac{1}{4}$ respectively. Find the probability of the problem being solved, if both of them try independently.

If $P\left(A\right)=\frac{3}{5}$ and $P\left(B\right)=\frac{1}{5}$ find $P(A\cap B),$ if $A$ and $B$ are independent events.

If $P\left(E\right)=0.6, P\left(F\right)=0.3$ and $P(E\cap F)=0.2$, then find $P\left(F|E\right)$.

If $P\left(A\right)=0.6, P\left(B\right)=0.3$ and $P(A\cap B)=0.2$ then find $P\left(\frac{A}{B}\right)$

If $A$ and $B$ are independent events such that odds in favour of $A$ is $2:3$ and odds against $B$ is then $4:5$, then $P\left(A\cap B\right)=$

The letters of the word "$\mathrm{LOGARITHM}$" are arranged at random. The probability that arrangements starts with vowel and end with consonant is

A man is known to speak the truth $2$ out of $3$ times. If he throws a die and reports that it is six, then the probability that it is actually five is

If $A$ and $B$ are two independent events such that $P\left(B\right)=\frac{2}{7}$ and $P\left(A\cup B^C\right)=0.8$, then $P(A\cup B)=$

When two dice are rolled, let $x$ be the probability of getting a sum of the numbers appear on the dice is at most $7$. Let $y$ be the probability of getting a sum $7$ at least once when a pair of dice are rolled $n$ times. In order to have $y>x,$ the minimum $n$ is

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

A coin is tossed three times, where event $E$: head on third toss, event $F$: heads on first two tosses. Determine $P\left(E|F\right)$.

Given that $\mathrm{E}$ and $\mathrm{F}$ are events such that $\mathrm{P}\left(\mathrm{E}\right)=0.6,\mathrm{P}\left(\mathrm{F}\right)=0.3$ and $\mathrm{P}\left(\mathrm{E}\cap \mathrm{F}\right)=0.2$, find $\mathrm{P}\left(\mathrm{E}|\mathrm{F}\right)$and $\mathrm{P}\left(\mathrm{F}|\mathrm{E}\right)$.