Two dice are thrown simultaneously to get the co-ordinates of a point on x-y plane. The probability that this point lies on or inside the region bounded by |x|+|y|=3.

Options

A. `1/3`
B. `1/4`
C. `1/12`
D. `1/36`

Answer

To solve the problem, we need to find the probability that the point (x, y) obtained by throwing two dice lies on or inside the region bounded by the equation |x| + |y| = 3.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- When two dice are thrown, the possible outcomes for each die are 1 to 6. Therefore, the coordinates (x, y) can take values from the set {1, 2, 3, 4, 5, 6} for both x and y.
- We need to find the points that satisfy the inequality |x| + |y| ≤ 3.

2. **Analyzing the Region**:
- The equation |x| + |y| = 3 describes a diamond (or rhombus) shape centered at the origin in the x-y plane.
- The vertices of this diamond are at the points (3, 0), (0, 3), (-3, 0), and (0, -3).
- However, since x and y must be positive integers (1 to 6), we only consider the first quadrant where both x and y are non-negative.

3. **Identifying Valid Points**:
- In the first quadrant, we need to find the integer points (x, y) such that x + y ≤ 3.
- The valid combinations of (x, y) that satisfy this condition are:
- (1, 1): 1 + 1 = 2
- (1, 2): 1 + 2 = 3
- (2, 1): 2 + 1 = 3
- (3, 0): 3 + 0 = 3 (not valid since y must be at least 1)
- (0, 3): 0 + 3 = 3 (not valid since x must be at least 1)

Thus, the valid points are:
- (1, 1)
- (1, 2)
- (2, 1)

4. **Counting Total Outcomes**:
- The total number of outcomes when throwing two dice is 6 (for the first die) × 6 (for the second die) = 36.

5. **Calculating the Probability**:
- The number of favorable outcomes (points that lie on or inside the region) is 3.
- Therefore, the probability P that the point lies on or inside the region is given by:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{3}{36} = \frac{1}{12} \]

### Final Answer:
The probability that the point lies on or inside the region bounded by |x| + |y| = 3 is \(\frac{1}{12}\).