Probability of Unique Solution in Binary Linear System

To determine the correctness of the given statements, we analyze the homogeneous system of linear equations:

$$\begin{cases} ax + by = 0 \\ cx + dy = 0 \end{cases}$$

where each coefficient $a, b, c, d \in \{0, 1\}$.

1. Analysis of STATEMENT-2:

The given system is a homogeneous system of linear equations.
For any values of $a, b, c, d \in \{0, 1\}$, the trivial solution $(x, y) = (0, 0)$ always satisfies both equations:

$$a(0) + b(0) = 0 \quad \text{and} \quad c(0) + d(0) = 0$$

Since a solution always exists for all possible choices of $(a, b, c, d)$, the probability that the system of equations has a solution is:

$$P(\text{system has a solution}) = 1$$

Therefore, STATEMENT-2 is True.

---

2. Analysis of STATEMENT-1:

The total number of possible ordered quadruplets $(a, b, c, d)$ is:

$$2 \times 2 \times 2 \times 2 = 2^4 = 16$$

A homogeneous system of two linear equations in two variables has a unique solution (the trivial solution) if and only if the determinant of its coefficient matrix is non-zero:

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

Since $a, b, c, d \in \{0, 1\}$, the products $ad$ and $bc$ can each only take the values $0$ or $1$. Thus, $\Delta \neq 0$ can occur in two disjoint cases:

1. Case 1: $ad = 1$ and $bc = 0$

   • $ad = 1 \implies a = 1$ and $d = 1$ (1 choice).
   • $bc = 0 \implies (b, c) \in \{(0, 0), (0, 1), (1, 0)\}$ (3 choices).
   • Number of outcomes $= 1 \times 3 = 3$.

2. Case 2: $ad = 0$ and $bc = 1$

   • $bc = 1 \implies b = 1$ and $c = 1$ (1 choice).
   • $ad = 0 \implies (a, d) \in \{(0, 0), (0, 1), (1, 0)\}$ (3 choices).
   • Number of outcomes $= 3 \times 1 = 3$.

Thus, the total number of favorable outcomes for a unique solution is:

$$n(E) = 3 + 3 = 6$$

The probability that the system has a unique solution is:

$$P(\text{unique solution}) = \frac{n(E)}{n(S)} = \frac{6}{16} = \frac{3}{8}$$

Therefore, STATEMENT-1 is True.

---

3. Relationship between the Statements:

STATEMENT-2 is a property of any homogeneous linear system (the existence of the trivial solution), but it does not account for the condition $\Delta \neq 0$ which determines whether the system has a unique solution or infinitely many solutions. Hence, STATEMENT-2 is not a correct explanation for STATEMENT-1.

Therefore, the correct choice is B.