An up-right path from lattice points $P$ and $Q$ on the $xy$-plane is a path in which every move is either one unit right or one unit up. The probability that a randomly chosen up-right path from $(0,0)$ to $(10,3)$ does not intersect the graph of $y=x^2+0.5$ can be written as $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

[i]Proposed by [b]HrishiP[/b][/i]
To solve this problem, we need to calculate the probability that a randomly chosen up-right path from \((0,0)\) to \((10,3)\) does not intersect the graph of \(y = x^2 + 0.5\).

1. **Calculate the total number of up-right paths from \((0,0)\) to \((10,3)\):**

   An up-right path from \((0,0)\) to \((10,3)\) consists of 10 right moves and 3 up moves. The total number of such paths is given by the binomial coefficient:
   \[
   \binom{13}{3} = \frac{13!}{3! \cdot 10!}
   \]

2. **Determine the points where the path might intersect \(y = x^2 + 0.5\):**

   The graph of \(y = x^2 + 0.5\) is a parabola. We need to check for integer points \((x, y)\) on the path where \(y \leq 3\). The relevant points are:
   \[
   \begin{aligned}
   &x = 0 \implies y = 0.5 \quad (\text{not an integer}) \\
   &x = 1 \implies y = 1.5 \quad (\text{not an integer}) \\
   &x = 2 \implies y = 4.5 \quad (\text{not an integer}) \\
   &x = 3 \implies y = 9.5 \quad (\text{not an integer})
   \end{aligned}
   \]
   Since none of these points are integers, the path does not intersect the graph of \(y = x^2 + 0.5\).

3. **Calculate the number of paths that do not intersect the graph:**

   Since the graph does not intersect any integer points within the range of our path, all paths from \((0,0)\) to \((10,3)\) are valid and do not intersect the graph.

4. **Calculate the probability:**

   Since all paths are valid, the probability is:
   \[
   \frac{\text{Number of valid paths}}{\text{Total number of paths}} = \frac{\binom{13}{3}}{\binom{13}{3}} = 1
   \]

5. **Express the probability in the form \(\frac{m}{n}\):**

   The probability is \(1\), which can be written as \(\frac{1}{1}\). Here, \(m = 1\) and \(n = 1\).

6. **Find \(m + n\):**

   \[
   m + n = 1 + 1 = 2
   \]

The final answer is \(\boxed{2}\).