In a group of $20$ boys and $10$ girls, $5$ boys and $3$ girls are honest. If two persons are selected at random from the group, find the probability that either both are girls or both are honest. Should there be gender discrimination? 

In the following experiment specify appropriate sample space.

A person is noting down the number of accidents along a busy highway during a year.

Two dice are thrown and the sum of the numbers which come up on the dice is noted. Let us consider the following events associated with this experiment

$\mathrm{A}:$ 'the sum is even'

$\mathrm{B}:$ 'the sum is a multiple of $3^{\text{'}}$

$\mathrm{C}:$ 'the sum is less than $4\text{'}$

$\mathrm{D}:$ 'the sum is greater than $11\text{'}$

Which pair of these events are mutually exclusive?

If sample space $\mathrm{S}=\left\{w_1,w_2,w_3,w_4,w_y,w_6,w_7\right]$, then which of the following assignments of probabilities are valid?

$\begin{array}{|cccccccc|}\hline \text{ Outcomes }& w_1& w_2& w_3& w_4& w_5& w_6& w_7\\ \left(i\right)& 0.1& 0.01& 0.05& 0.03& 0.01& 0.2& 0.6\\ \left(ii\right)& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}\\ \left(iii\right)& 0.1& 0.2& 0.3& 0.4& 0.5& 0.6& 0.7\\ \left(iv\right)& -0.1& 0.2& 0.3& 0.4& -0.2& 0.1& 0.3\\ \left(v\right)& \frac{1}{14}& \frac{2}{14}& \frac{3}{14}& \frac{4}{14}& \frac{5}{14}& \frac{6}{14}& \frac{15}{14}\\ \hline\end{array}$