In fencing, you win a round if you are the first to reach $15$ points. Suppose that when $A$ plays against $B$, at any point during the round, $A$ scores the next point with probability $p$ and $B$ scores the next point with probability $q=1-p$. (However, they never can both score a point at the same time.)

Suppose that in this round, $A$ already has $14-k$ points, and $B$ has $14-\ell$ (where $0\le k,\ell\le 14$). By how much will the probability that $A$ wins the round increase if $A$ scores the next point?
To solve this problem, we need to determine how the probability that player \( A \) wins the round changes if \( A \) scores the next point. We start by defining the current scores and the probabilities involved.

1. **Define the current scores and probabilities:**
   - \( A \) has \( 14 - k \) points.
   - \( B \) has \( 14 - \ell \) points.
   - \( A \) scores the next point with probability \( p \).
   - \( B \) scores the next point with probability \( q = 1 - p \).

2. **Determine the probability that \( A \) wins the round:**
   - \( A \) needs \( k + 1 \) more points to reach 15 points and win.
   - \( B \) can score at most \( \ell \) more points before \( A \) reaches 15 points.

3. **Calculate the probability that \( A \) wins if \( A \) scores the next point:**
   - If \( A \) scores the next point, \( A \) will have \( 15 - k \) points, needing \( k \) more points to win.
   - The probability that \( A \) wins from this new state is given by:
     \[
     \sum_{i=0}^{\ell} \binom{k-1+i}{k-1} p^k q^i
     \]

4. **Calculate the probability that \( A \) wins if \( B \) scores the next point:**
   - If \( B \) scores the next point, \( B \) will have \( 15 - \ell \) points, needing \( \ell \) more points to win.
   - The probability that \( A \) wins from this new state is given by:
     \[
     \sum_{i=0}^{\ell} \binom{k+i}{k} p^{k+1} q^i
     \]

5. **Determine the difference in probabilities:**
   - The difference in the probability that \( A \) wins if \( A \) scores the next point versus if \( B \) scores the next point is:
     \[
     p^k \sum_{i=0}^{\ell} \left[ \binom{k-1+i}{k-1} q^i - \binom{k+i}{k} p q^i \right]
     \]

6. **Simplify the expression:**
   - Using the binomial coefficient identity \(\binom{n}{k+1} = \binom{n-1}{k+1} + \binom{n-1}{k}\), we can simplify the expression:
     \[
     \begin{aligned}
     A &= \sum_{i=0}^{\ell} q^i \left[ \binom{k-1+i}{k-1} - \binom{k+i}{k} \right] + \sum_{i=0}^{\ell} \binom{k+i}{k} q^{i+1} \\
     &= -\sum_{i=1}^{\ell} \binom{k-1+i}{k} q^i + \sum_{i=0}^{\ell} \binom{k+i}{k} q^{i+1} \\
     &= \binom{k+\ell}{k} q^{\ell+1}
     \end{aligned}
     \]

7. **Final probability increase:**
   - The increase in the probability that \( A \) wins the round if \( A \) scores the next point is:
     \[
     p^k \cdot \binom{k+\ell}{k} q^{\ell+1}
     \]

The final answer is \(\boxed{\binom{k+\ell}{k} q^{\ell+1} p^k}\)