If Aand B are two events than the value of the determinant choosen at random from all the determinants of order 2 with entries 0 or 1 only is positive or negative respectively. Then (a) `P(A) ge P(B)` (b) `P(A) le P(B)` (c) `P(A)= P(B)=1/2` (d)None of these

Options

A. `P(A) gt P(B)`
B. `P(A) lt P(B)`
C. `P(A)= P(B) = 1//2`
D. `None of these

Answer

To solve the problem, we need to analyze the determinants of order 2 with entries 0 or 1, and determine the probabilities of events A and B based on the properties of these determinants.

### Step-by-Step Solution:

1. **Understanding the Determinant of a 2x2 Matrix**:
The determinant of a 2x2 matrix is given by:
\[ \text{det} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc \]
where \( a, b, c, d \) can be either 0 or 1.

2. **Total Possible Matrices**:
Since each entry can be either 0 or 1, the total number of 2x2 matrices is:
\[ 2^4 = 16 \]
This is because there are 4 entries, and each can take 2 values.

3. **Counting Positive Determinants (Event A)**:
We need to find the cases where \( ad - bc > 0 \) (positive determinant):
- If \( a = 1 \) and \( d = 1 \), then \( ad = 1 \). We need \( bc < 1 \) (i.e., either \( b = 0 \) or \( c = 0 \)).
- Possible combinations:
- \( b = 0, c = 0 \) → \( (1, 0, 0, 1) \)
- \( b = 0, c = 1 \) → \( (1, 0, 1, 1) \)
- \( b = 1, c = 0 \) → \( (1, 1, 0, 1) \)

Thus, the positive determinant cases are:
- \( (1, 0, 0, 1) \)
- \( (1, 0, 1, 1) \)
- \( (1, 1, 0, 1) \)

Total positive cases = 3.

4. **Counting Negative Determinants (Event B)**:
Now we find the cases where \( ad - bc < 0 \) (negative determinant):
- If \( bc > ad \), we can have:
- \( b = 1, c = 1 \) → \( bc = 1 \) and \( ad \) must be 0.
- Possible combinations:
- \( a = 0, d = 0 \) → \( (0, 1, 1, 0) \)
- \( a = 0, d = 1 \) → \( (0, 1, 1, 1) \)
- \( a = 1, d = 0 \) → \( (1, 1, 1, 0) \)

Thus, the negative determinant cases are:
- \( (0, 1, 1, 0) \)
- \( (0, 1, 1, 1) \)
- \( (1, 1, 1, 0) \)

Total negative cases = 3.

5. **Calculating Probabilities**:
The probabilities of events A and B are:
\[ P(A) = \frac{\text{Number of positive cases}}{\text{Total cases}} = \frac{3}{16} \]
\[ P(B) = \frac{\text{Number of negative cases}}{\text{Total cases}} = \frac{3}{16} \]

6. **Conclusion**:
Since \( P(A) = P(B) = \frac{3}{16} \), we can conclude that:
- The correct option is (d) None of these, as none of the other options are true.