Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled $0, 1, \dots, 9$. First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll?
1. **Define the problem and variables:**
   - Let the three dice be \( D_1, D_2, \) and \( D_3 \).
   - The faces of each die are labeled \( 0, 1, 2, \ldots, 9 \).
   - We need to find the probability that the sum of the first two dice \( D_1 + D_2 \) equals the third die \( D_3 \).

2. **Calculate the possible outcomes:**
   - Each die has 10 faces, so there are \( 10 \times 10 \times 10 = 1000 \) possible outcomes for the three dice rolls.

3. **Determine the range of sums for \( D_1 + D_2 \):**
   - The minimum sum of \( D_1 + D_2 \) is \( 0 + 0 = 0 \).
   - The maximum sum of \( D_1 + D_2 \) is \( 9 + 9 = 18 \).

4. **Count the favorable outcomes:**
   - For each possible value of \( D_3 \) (ranging from 0 to 9), we need to count the number of pairs \((D_1, D_2)\) such that \( D_1 + D_2 = D_3 \).
   - For \( D_3 = k \) where \( 0 \leq k \leq 9 \), the number of pairs \((D_1, D_2)\) that sum to \( k \) is \( k + 1 \) (since \( D_1 \) can be any value from 0 to \( k \), and \( D_2 = k - D_1 \)).
   - For \( D_3 = k \) where \( 10 \leq k \leq 18 \), the number of pairs \((D_1, D_2)\) that sum to \( k \) is \( 19 - k \) (since \( D_1 \) can be any value from \( k - 9 \) to 9, and \( D_2 = k - D_1 \)).

5. **Sum the favorable outcomes:**
   - For \( D_3 = 0 \) to \( 9 \):
     \[
     \sum_{k=0}^{9} (k + 1) = 1 + 2 + 3 + \ldots + 10 = \frac{10 \times 11}{2} = 55
     \]
   - For \( D_3 = 10 \) to \( 18 \):
     \[
     \sum_{k=10}^{18} (19 - k) = 9 + 8 + 7 + \ldots + 1 = \frac{9 \times 10}{2} = 45
     \]

6. **Total number of favorable outcomes:**
   \[
   55 + 45 = 100
   \]

7. **Calculate the probability:**
   - The probability is the number of favorable outcomes divided by the total number of possible outcomes:
     \[
     \frac{100}{1000} = \frac{1}{10}
     \]

The final answer is \(\boxed{\frac{1}{10}}\)