Probability Questions

Let $X$ and $Y$ be two independent Binomial random variables such that $X\sim B(2,p)$ and $Y\sim B(4,p)$. If $P(X\ge 1)=5/9$, find (i) $P(Y=4)$ and (ii) $V(2X-Y)$.

A die is thrown 3 times. Let Event A be "4 on the 3rd throw" and Event B be "6 on the 1st and 5 on the 2nd throw". What is the probability of A given that B has already occurred?

If an event $E$ has only one sample point of a sample space, it is called a ______:

A player tosses three coins. If the first toss is a Head, find the probability that there are two consecutive Heads among the three coins.

A family has two children. If the first child is a boy, find the probability that the other child is also a boy.