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 coordinates obtained from throwing two dice lie 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 outcomes can be represented as points (x, y) on the x-y plane, where x is the result of the first die and y is the result of the second die. Each die has 6 faces, so the possible outcomes are (1,1), (1,2), ..., (6,6).

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

3. **Understanding the Equation |x| + |y| = 3**:
- The equation |x| + |y| = 3 describes a diamond shape (or rhombus) on the x-y plane with vertices at (3,0), (0,3), (-3,0), and (0,-3). We need to find the integer points (x, y) that lie on or inside this shape.

4. **Finding the Points**:
- The points (x, y) that satisfy |x| + |y| ≤ 3 can be found by considering the combinations of x and y values from 1 to 6 (the faces of the dice). We need to check which combinations satisfy the inequality.

5. **Listing the Valid Points**:
- The valid combinations that lie on or inside the diamond are:
- (3,0)
- (0,3)
- (2,1)
- (1,2)
- (1,1)
- (2,2)
- (0,0)
- (1,0)
- (0,1)
- (0,-1)
- (-1,0)
- (-1,-1)
- (-2,0)
- (0,-2)
- (-2,1)
- (-1,2)
- (2,-1)
- (1,-2)
- However, since x and y can only be from 1 to 6, we only consider the positive coordinates:
- (1, 1), (1, 2), (2, 1), (2, 2), (3, 0), (0, 3)

6. **Counting the Favorable Outcomes**:
- The valid points from the dice outcomes that satisfy |x| + |y| ≤ 3 are:
- (1, 1)
- (1, 2)
- (2, 1)
- (3, 0)
- (0, 3)
- Thus, the favorable outcomes are (1,1), (1,2), (2,1), (2,2), (3,0), and (0,3).
- However, we must only consider points that can be achieved with the dice, which are (1,1), (1,2), (2,1), and (2,2).
- Therefore, the total number of favorable outcomes is 5.

7. **Calculating the Probability**:
- The probability is given by the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{5}{36} \]

### Final Answer:
The probability that the point lies on or inside the region bounded by |x| + |y| = 3 is \( \frac{5}{36} \).