To find the probability that three dice show numbers in arithmetic progression (A.P.), we can follow these steps:
### Step 1: Determine the Total Outcomes
When three dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing three dice is given by:
\[
\text{Total Outcomes} = 6^3 = 216
\]
### Step 2: Understand the Condition for A.P.
For three numbers \(A\), \(B\), and \(C\) to be in A.P., they must satisfy the condition:
\[
2B = A + C
\]
This means that \(B\) is the average of \(A\) and \(C\).
### Step 3: Identify Possible Values for \(B\)
The middle number \(B\) can take values from 1 to 6. We will analyze each possible value of \(B\) to find corresponding pairs \((A, C)\) that satisfy the A.P. condition.
### Step 4: Calculate Valid Combinations for Each Value of \(B\)
1. **For \(B = 1\)**:
- \(A + C = 2 \times 1 = 2\)
- Possible pairs: \((1, 1)\) → 1 combination.
2. **For \(B = 2\)**:
- \(A + C = 2 \times 2 = 4\)
- Possible pairs: \((1, 3), (2, 2), (3, 1)\) → 3 combinations.
3. **For \(B = 3\)**:
- \(A + C = 2 \times 3 = 6\)
- Possible pairs: \((1, 5), (2, 4), (3, 3), (4, 2), (5, 1)\) → 5 combinations.
4. **For \(B = 4\)**:
- \(A + C = 2 \times 4 = 8\)
- Possible pairs: \((2, 6), (3, 5), (4, 4), (5, 3), (6, 2)\) → 5 combinations.
5. **For \(B = 5\)**:
- \(A + C = 2 \times 5 = 10\)
- Possible pairs: \((4, 6), (5, 5), (6, 4)\) → 3 combinations.
6. **For \(B = 6\)**:
- \(A + C = 2 \times 6 = 12\)
- Possible pairs: \((6, 6)\) → 1 combination.
### Step 5: Total Valid Combinations
Now, we sum the valid combinations:
\[
1 + 3 + 5 + 5 + 3 + 1 = 18
\]
Thus, there are 18 favorable outcomes where the numbers on the dice are in A.P.
### Step 6: Calculate the Probability
The probability \(P\) that the three dice show numbers in A.P. is given by:
\[
P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{18}{216} = \frac{1}{12}
\]
### Conclusion
The probability that three dice show numbers in A.P. is:
\[
\boxed{\frac{1}{12}}
\]
