Five positive integers from $1$ to $15$ are chosen without replacement.  What is the probability that their sum is divisible by $3$?
1. **Identify the residues modulo 3:**
   The numbers from 1 to 15 can be categorized based on their residues modulo 3:
   \[
   \begin{aligned}
   &\text{Residue } 0 \mod 3: & 3, 6, 9, 12, 15 \\
   &\text{Residue } 1 \mod 3: & 1, 4, 7, 10, 13 \\
   &\text{Residue } 2 \mod 3: & 2, 5, 8, 11, 14 \\
   \end{aligned}
   \]
   Each residue class has exactly 5 numbers.

2. **Total number of ways to choose 5 numbers from 15:**
   The total number of ways to choose 5 numbers out of 15 is given by the binomial coefficient:
   \[
   \binom{15}{5}
   \]

3. **Sum of residues modulo 3:**
   When choosing 5 numbers, the sum of their residues modulo 3 can be 0, 1, or 2. We need to find the probability that the sum is 0 modulo 3.

4. **Equal probability of each residue sum:**
   Since there are equal numbers of each residue (5 of each), the probability that the sum of the chosen numbers is 0 modulo 3 is the same as the probability that it is 1 modulo 3 or 2 modulo 3. This is due to the symmetry and uniform distribution of residues.

5. **Calculate the probability:**
   Since the sum of the residues modulo 3 can be 0, 1, or 2 with equal probability, the probability that the sum is 0 modulo 3 is:
   \[
   \frac{1}{3}
   \]

\(\blacksquare\)

The final answer is \( \boxed{ \frac{1}{3} } \)