Probability

A signal which can be green or red with probability $\dfrac45$ and $\dfrac15$ respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is $\dfrac34$. If the signal received at station B is green, then the probability that the original signal was green is

The probability that $X=3$ equals

The probability that $X\ge3$ equals

The conditional probability that $X\ge6$ given $X>3$ equals

Consider the system of equations $ax+by=0$, $cx+dy=0$, where $a,b,c,d\in\{0,1\}$.

STATEMENT-1: The probability that the system of equations has a unique solution is $\dfrac38$.

and

STATEMENT-2: The probability that the system of equations has a solution is 1.

An experiment has 10 equally likely outcomes. Let $A$ and $B$ be two non-empty events of the experiment. If $A$ consists of 4 outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

Let $H_1,H_2,\ldots,H_n$ be mutually exclusive and exhaustive events with $P(H_i)>0$, $i=1,2,\ldots,n$. Let $E$ be any other event with $0<P(E)<1$.

STATEMENT-1: $P(H_i\mid E)>P(E\mid H_i)\cdot P(H_i)$ for $i=1,2,\ldots,n$.

because

STATEMENT-2: $\displaystyle\sum_{i=1}^nP(H_i)=1$.

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