Conditional probability

If A and B are any two events such that $P(A)=\frac25$ and $P(A\cap B)=\frac3{20}$, then the conditional probability, $P(A\mid(A'\cup B'))$, where A′ denotes the complement of A, is equal to :

One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II is $\dfrac13$, then the correct option(s) with the possible values of $n_1$, $n_2$, $n_3$ and $n_4$ is(are)

A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is $\dfrac13$, then the correct option(s) with the possible values of $n_1$ and $n_2$ is(are)

If the lengths of the sides of a triangle are decided by the three throws of a single fair die, then the probability that the triangle is of maximum area given that it is an isosceles triangle, is :

Let A and E be any two events with positive probabilities :

Statement - 1 : $P(E/A)\ge P(A/E)P(E)$

Statement - 2 : $P(A/E)\ge P(A\cap E)$.

A ship is fitted with three engines $E_1$, $E_2$ and $E_3$. The engines function independently of each other with respective probabilities $\frac12$, $\frac14$ and $\frac14$. For the ship to be operational at least two of its engines must function. Let $X$ denote the event that the ship is operational and let $X_1$, $X_2$ and $X_3$ denote respectively the events that the engines $E_1$, $E_2$ and $E_3$ are functioning. Which of the following is (are) true?

Let $X$ and $Y$ be two events such that $P(X|Y)=\dfrac12$, $P(Y|X)=\dfrac13$ and $P(X\cap Y)=\dfrac16$. Which of the following is (are) correct?

The probability of the drawn ball from $U_2$ being white is

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

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 $E^c$ 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