If one horse is given 1:50 odds but should have been given 1:100, how many races would you have to bet on to make money on those odds?

Let's assume the odds are fair such that the odds of all horses sum up to 1. Betting on all other horses would give you 1:1.02 odds whereas they should have been 1:1.01. While there is no way to answer for sure how many races you would have to bet on to make money, we can look at the probabilities. There's a 95% likelihood that the horse will lose 5 races in a row. If you start with $100 and reinvest your winnings every time you'd earn a profit of $10 with 95% likelihood, meaning there's a positive expectation value. On the other hand, if the 1:50 odds were right, the probability of it losing 5 times in a row would be around 90%, leading to a neutral expectation value.

Except investing isn't like a horse race. Each horse has a different finish line, and those finish lines are in flux during the race. Crucially, you can change your bet in the middle of the race.