Consider the system of equations

$ax + by = 0$

$cx + dy = 0 $

where a, b, c, d, ∈{0, 1}.

Statement-1: The probability that the system of equations has a unique solution is $\frac{3}{8}.$

Statement-2: The probability that the system has  a solution is 1.

Statement 1 is True, Statement 2 is True; Statement 2 is not a correct explanation for Statement 1.

The given system of equations is a homogenous system of equations which is always consistent. SO, the probability that the system has a solution is 1.

Hence, statement-2 is true.

$\begin{vmatrix} a&b\\c&d\end{vmatrix}= ad - bc ≠ 0.$

As a, b, c, d ∈ {0,1}. So, each of a, b, c and d can assume two values. Therefore, there are 2⁴ sets of values of a, b, c and d.

Clearly, ad -bc ≠ 0 iff ad = 1 and bc = 0 or ad = 0 and bc = 1

Now, ad = 1 and bc = 0 iff (a =1, d =1 and b = 1, c=0)

or (a = 1, d=1 and b = 0, ,c= 1) or a = 1, d = 1 and b = 0, c= 0).

So, there are three sets of values of a, b, c, d satisfying ad = 1 and bc = 0.

Similarly, there are three sets of values of a, b, c, d satisfying bc = 1 and ad = 0

Thus, out of 2⁴ sets of values of a, b, c and d, Therefore, six sets for which the given system has a unique solution.

∴ Probability that the system has a unique solution $=\frac{6}{16}=\frac{3}{8}$

Hence, statement-1 is true. But, statement-2 is not a correct explanation for statement-1.