Question about odds vs. percentages, inspired by Arieh

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Percentages and odds are just different ways of expressing the same thing.

If you have a 10% chance to hit your draw, there is a 90% chance you won't. So, the odds are 9:1 against.

In poker, most people use the size of the pot compared to the size of the bet to get pot odds. So, if there is $30 in the pot, and $3 to call, your pot odds are 10:1.

If the pot odds are greater than the odds to hit your hand (i.e. 10:1 pot odds on $3 bet vs. 9:1 odds to hit your 10% draw), then you can call.

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Is there a simple way to compute odds into a percentage? I see the 9:1 and 10:1 examples easily, but what about 3:1. It's not just 1 (divided by) 3, (for 33% to win) is it? Basically I'm just looking for the formula here.

Sorry, I still feel as if I should know this. Something about it just isn't sticking. And why did Arieh's learning this concept mean he can now play more hands?