The Tigers beat the Sharks $2$ out of $3$ times they played. They then played $N$ more times, and the Sharks ended up winning at least $95\%$ of all the games played. What is the minimum possible value for $N$?
In order for $N$ to be the smallest, the Sharks must win all the subsequent games after the initial three games. This gives us $$\frac{1+N}{3+N}\ge \frac{95}{100}\implies N=\boxed{37}$$