Solution

Mean $(\bar{x}) = 8$ (Given)

$\Rightarrow \frac{2 + 4 + 10 + x + 12 + 14 + y}{7} = 8$

$\Rightarrow x + y = 14 \quad ...(1)$

Variance $(\sigma^2) = 16$ (Given)

$\Rightarrow 16 = \frac{2^2 + 4^2 + 10^2 + x^2 + 12^2 + 14^2 + y^2}{7} - 8^2$

$\Rightarrow x^2 + y^2 = 100 \quad ...(2)$

$(x + y)^2 = x^2 + y^2 + 2xy$

$\Rightarrow 14^2 = 100 + 2xy \Rightarrow xy = 48$

Since $x > y$

$\Rightarrow x = 8,; y = 6$

Now set $X = {1,2,3,4,6,5}$

Now we choose two numbers one after another without replacement

Total outcomes $= 6 \times 5 = 30$

We want the probability that smaller number $< 4$

$P(\text{smaller} < 4) = 1 - P(\text{smaller} \ge 4)$

$= 1 - \frac{6}{30} = \frac{4}{5}$