A cube with all six faces coloured is cut into $64$ cubical blocks of the same size which are thoroughly mixed. Find the probability that the $2$ randomly chosen blocks have $2$ coloured faces each.

A player tosses an unbiased coin and is to score two points for every head turned up and one point for every tail turned up. If $P_n$ denotes the probability that his score is exactly $n$ points, prove that

$P_n-P_{n-1}=\frac{1}{2}\left(P_{n-2}-P_{n-1}\right),n\ge 3$

The probabilities that three men hit a target are, respectively, $0.3,0.5$ and $0.4$. Each fires once at the target. (As usual, assume that the three events that each hits the target are independent)

$\left(a\right)$ Find the probability that they all : (i) hit the target ; (ii) miss the target

$\left(b\right)$ Find the probability that the target is hit: (i) atleast once, (ii) exactly once.

$\left(c\right)$ If only one hits the target, what is the probability that is was the first man?

Let $A$ & $B$ two events defined on a sample space. Given $P\left(A\right)=0.4;P\left(B\right)=0.80$ and $P\left(\frac{\overline{A}}{\overline{B}}\right)=0.10$. Then find :

 $\left(\mathrm{i}\right) P\left(\overline{A}\cup B\right)$

 $\left(\mathrm{ii}\right) P\left[\left(\overline{A}\cap B\right)\cup \left(A\cap \overline{B}\right)\right]$