Step-by-step Solution:

1. Analyze the System of Equations and Sample Space

The given system of equations is: $ax + by = 0$ $cx + dy = 0$ This is a system of two homogeneous linear equations in two variables, x and y. The coefficients $a, b, c, d$ are chosen from the set $\{0, 1\}$. Since each of the four coefficients can take 2 possible values (0 or 1), the total number of possible systems of equations is $2 \times 2 \times 2 \times 2 = 2^4 = 16$. This is the size of our sample space.

2. Evaluate STATEMENT - 1

STATEMENT - 1 says: "The probability that the system of equations has a unique solution is $3/8$."

A system of homogeneous linear equations has a unique solution (the trivial solution $x=0, y=0$) if and only if the determinant of the coefficient matrix is non-zero. The coefficient matrix is $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. The condition for a unique solution is $\det(A) = ad - bc \neq 0$.

We need to find the number of combinations of $(a, b, c, d)$ from $\{0, 1\}^4$ for which $ad - bc \neq 0$. Since $a, b, c, d \in \{0, 1\}$, the products $ad$ and $bc$ can only be 0 or 1.

For $ad - bc \neq 0$, we have two possible cases:

• Case 1: $ad - bc = 1$ This implies $ad=1$ and $bc=0$.

  • For $ad=1$, we must have $a=1$ and $d=1$. This is 1 possibility for the pair $(a,d)$.
  • For $bc=0$, at least one of $b$ or $c$ must be 0. The possible pairs for $(b,c)$ are $(0,0), (0,1), (1,0)$. This gives 3 possibilities.
  • The total number of systems in this case is $1 \times 3 = 3$.

• Case 2: $ad - bc = -1$ This implies $ad=0$ and $bc=1$.

  • For $bc=1$, we must have $b=1$ and $c=1$. This is 1 possibility for the pair $(b,c)$.
  • For $ad=0$, at least one of $a$ or $d$ must be 0. The possible pairs for $(a,d)$ are $(0,0), (0,1), (1,0)$. This gives 3 possibilities.
  • The total number of systems in this case is $3 \times 1 = 3$.

The total number of favorable outcomes for a unique solution is the sum of outcomes from both cases: $3 + 3 = 6$.

The probability of the system having a unique solution is: $P(\text{unique solution}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{6}{16} = \frac{3}{8}$ Therefore, STATEMENT - 1 is True.

3. Evaluate STATEMENT - 2

STATEMENT - 2 says: "The probability that the system of equations has a solution is 1."

The given system is a homogeneous system of linear equations. A homogeneous system always has at least one solution, which is the trivial solution $(x,y) = (0,0)$. We can verify this by substituting $x=0$ and $y=0$ into the equations: $a(0) + b(0) = 0$ $c(0) + d(0) = 0$ These equations hold true for any values of $a, b, c, d$. Since the system always has a solution for all 16 possible combinations of coefficients, the event "the system has a solution" is a certain event. The probability of a certain event is 1. Therefore, STATEMENT - 2 is True.

4. Analyze the Relationship Between the Statements

We have determined that both STATEMENT - 1 and STATEMENT - 2 are true. Now, we must check if STATEMENT - 2 is the correct explanation for STATEMENT - 1.

• STATEMENT - 1 calculates the probability of a unique solution. The value $3/8$ is derived from counting the specific combinations of coefficients where the determinant $ad-bc$ is non-zero.
• STATEMENT - 2 states a general property of homogeneous systems: they are always consistent (i.e., they always have at least one solution). The system can have either a unique solution or infinitely many solutions, but it never has no solution.

The fact that the system always has a solution (STATEMENT - 2) is a necessary precondition, but it does not explain why the probability of the solution being unique is specifically $3/8$. The reason for this probability is the combinatorial result from step 2. Therefore, STATEMENT - 2 is not the correct explanation for STATEMENT - 1.

Conclusion:

STATEMENT - 1 is True, STATEMENT - 2 is True; but STATEMENT - 2 is NOT a correct explanation for STATEMENT - 1. This corresponds to option B.