Answer: 116

Solution:

All $\binom{10}{4} = 210$ draws are equally likely. The number of draws sharing exactly $k$ numbers with Jen's ticket is $\binom{4}{k}\binom{6}{4-k},$ so the number winning a prize is $$\begin{aligned} &\binom{4}{2}\binom{6}{2} + \binom{4}{3}\binom{6}{1} \\ &\quad {}+ \binom{4}{4}\binom{6}{0} \\ &= 90 + 24 + 1 = 115, \end{aligned}$$ and exactly $1$ of these wins the grand prize.

Since the grand prize implies a prize, the conditional probability is $\frac{1/210}{115/210} = \frac{1}{115},$ so $m + n = 1 + 115 = 116.$