Independent events

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?

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\dfrac{11}{25}$ and the probability of none of them occurring is $\dfrac2{25}$. If $P(T)$ denotes the probability of occurrence of the event $T$, then

The probability that $X=3$ equals

The probability that $X\ge3$ equals

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

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

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