jmc

Positive integers which are congruent to $2(\mathrm{mod}5)$ belong to the set $$\{2+5(0),2+5(1),2+5(2),...,\}.$$To find the largest element of this set which is less than or equal to 50, we look for the largest possible integer $n$ such that $$2+5(n-1)\le 50.$$Solving this inequality, we find $n\le 53/5$, so the maximum integer solution is $n=\lfloor 53/5\rfloor =10$. Since there are 50 total tiles, the probability that the tile is marked with a number congruent to $2(\mathrm{mod}5)$ is $\frac{10\text{blue tiles}}{50\text{total tiles}}=\boxed{\frac{1}{5}}.$