Find the probability that at least one of poker opponents hits a card

Question

In Texas hold'em, when you see the flop there is about 33% chance that one of your 2 cards hits something on the flop, maybe not exactly but let's say it's 33% that your opponent did hit something on the flop with his 2 pocket cards. Let's say you have 2 opponents with you in the pot, then it should be 66% that one of them hit something, 3 opponents 99%? Well maybe but when there are 4 more people after you then it can't be 132% that doesn't make sense, there is no 100%, how do I calculate that correctly?

Answer 1

An approximation to the probability of at least one of them hitting would be 100% minus the probability of them both NOT hitting. Or $$1-\frac{2}{3}\cdot\frac{2}{3}=\frac{5}{9}$$ The probability of at least one of THREE players hitting would be $$1-\frac{2}{3}\cdot\frac{2}{3}\cdot\frac{2}{3}=\frac{19}{27}$$

This doesn't work exactly this way because of the way each players hand affects each other, (shared cards, etc.) but is an approximation.