Question

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

$$\Rightarrow \frac{2 + 4 + 10 + x + 12 + 14 + y}{7} = 8 $$$\Rightarrow x + y = 14\ldots.(1)\ldots$.

Variance $\left( \sigma^{2} \right) = 16$ (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\ \ldots..(2)\ldots.$.

$$\because(x + y)^{2} = x^{2} + y^{2} + 2xy $$$\Rightarrow xy = 48$ (sum is 14 product is 48 )

Since problem states $x > y$

$\therefore x = 8$ and $y = 6$

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

Now we choose two numbers one after author without replacement total outcomes $= 6 \times 5 = 30$

We want the prob. That the smaller number among the two is less than 4

$$\begin{matrix} P(\text{~}\text{smaller}\text{~} < 4) & \ = 1 - P(\text{~}\text{smaller}\text{~} \geq 4) \\ & \ = 1 - \frac{6}{30} = \frac{4}{5} \end{matrix}$$