1. Define the random variables:
Let $X$ be the number of ducks hit by Ali, and $Y$ be the number of ducks hit by Veli. Both $X$ and $Y$ follow a binomial distribution:
$$X \sim \text{Binomial}(12, \frac{1}{2})$$
$$Y \sim \text{Binomial}(13, \frac{1}{2})$$

2. Calculate the expected values and variances:
The expected value and variance for a binomial distribution $\text{Binomial}(n, p)$ are given by:
$$E[X] = np \quad \text{and} \quad \text{Var}(X) = np(1-p)$$
For Ali:
$$E[X] = 12 \cdot \frac{1}{2} = 6 \quad \text{and} \quad \text{Var}(X) = 12 \cdot \frac{1}{2} \cdot \frac{1}{2} = 3$$
For Veli:
$$E[Y] = 13 \cdot \frac{1}{2} = 6.5 \quad \text{and} \quad \text{Var}(Y) = 13 \cdot \frac{1}{2} \cdot \frac{1}{2} = 3.25$$

3. Standardize the variables:
To find the probability that $Y > X$, we can standardize $X$ and $Y$ to approximate them using the normal distribution (Central Limit Theorem):
$$Z_X = \frac{X - 6}{\sqrt{3}} \quad \text{and} \quad Z_Y = \frac{Y - 6.5}{\sqrt{3.25}}$$

4. Calculate the difference of the standardized variables:
Let $Z = Z_Y - Z_X$. Since $Z_X$ and $Z_Y$ are approximately standard normal variables, $Z$ will also be approximately normal with:
$$E[Z] = E[Z_Y] - E[Z_X] = 0$$
$$\text{Var}(Z) = \text{Var}(Z_Y) + \text{Var}(Z_X) = 1 + 1 = 2$$
Thus, $Z \sim N(0, 2)$.

5. Find the probability:
We need to find $P(Y > X)$, which is equivalent to $P(Z > 0)$. Since $Z \sim N(0, 2)$, we standardize $Z$ to a standard normal variable $Z'$:
$$Z' = \frac{Z}{\sqrt{2}} \sim N(0, 1)$$
Therefore,
$$P(Z > 0) = P\left(\frac{Z}{\sqrt{2}} > 0\right) = P(Z' > 0) = 0.5$$

Conclusion:
$$\boxed{\frac{1}{2}}$$