Alice picks a number uniformly at random from the first $5$ even positive integers, and Palice picks a number uniformly at random from the first $5$ odd positive integers. If Alice picks a larger number than Palice with probability $\frac{m}{n}$ for relatively prime positive integers $m,n$, compute $m+n$.

[i]2020 CCA Math Bonanza Lightning Round #4.1[/i]
1. **Identify the sets of numbers Alice and Palice can pick from:**
   - Alice picks from the first 5 even positive integers: \(\{2, 4, 6, 8, 10\}\).
   - Palice picks from the first 5 odd positive integers: \(\{1, 3, 5, 7, 9\}\).

2. **Calculate the total number of possible outcomes:**
   - Each player has 5 choices, so there are \(5 \times 5 = 25\) possible outcomes.

3. **Determine the winning conditions for Alice:**
   - Alice wins if her number is greater than Palice's number.
   - We need to count the number of favorable outcomes where Alice's number is greater than Palice's number.

4. **List the favorable outcomes:**
   - If Alice picks 2: No favorable outcomes (since 2 is less than all odd numbers in Palice's set).
   - If Alice picks 4: Favorable outcomes are (4, 1) and (4, 3) (2 outcomes).
   - If Alice picks 6: Favorable outcomes are (6, 1), (6, 3), and (6, 5) (3 outcomes).
   - If Alice picks 8: Favorable outcomes are (8, 1), (8, 3), (8, 5), and (8, 7) (4 outcomes).
   - If Alice picks 10: Favorable outcomes are (10, 1), (10, 3), (10, 5), (10, 7), and (10, 9) (5 outcomes).

5. **Count the total number of favorable outcomes:**
   \[
   0 + 2 + 3 + 4 + 5 = 14
   \]

6. **Calculate the probability that Alice picks a larger number than Palice:**
   \[
   \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{14}{25}
   \]

7. **Simplify the fraction \(\frac{14}{25}\):**
   - The fraction \(\frac{14}{25}\) is already in its simplest form since 14 and 25 are relatively prime.

8. **Compute \(m + n\) where \(\frac{m}{n} = \frac{14}{25}\):**
   \[
   m = 14, \quad n = 25 \quad \Rightarrow \quad m + n = 14 + 25 = 39
   \]

The final answer is \(\boxed{39}\)