Pot Odds vs. Percentage Change to Win

I'm a math professor. And the following misconception happens all the time. Someone uses math or PokerStove to calculate that (with only the river card to come) they have a 25% chance of hitting their hand and winning the pot. They say they need pot odds of 4 to 1 to be able to call. And they're incorrect. They only need 3 to 1 pot odds to call.

Odds are quoted as a win-loss ratio. Being ahead "3 to 2" means that if the event happened 5 times, Hero would win 3 of the 5 and lose 2 times. This is a 60% chance of winning.

Percentage Chance of Winning (aka "equity") is quoted as a Wins / Total fraction (converted to percentage form). If Hero will hit his outs and win the hand 25% of the time, this means if the event happened 100 times, Hero would win 25 times and lose 75 times (a 1 to 3 ratio).

By the way, poker odds are often quoted backwards, in a loss-win ratio, as in Hero has odds of 7.5 to 1 against hitting a set on the flop. To convert this to a percentage, we have to add both sides of the ratio together the denominator of the fraction:

% Chance of Flopping Set = 1 / (7.5 + 1 ) = 1 / 8.5 = .112 = 11.2% .

Converting equity or % Chance to Win to odds format is more difficult unless the percent is divides 100 evenly, like 10%.

If Hero has 10% chance of winning, his odds are 9 to 1 against.
If Hero has 75% chance of winning, his odds are 3 to 1.
If Hero has 20% chance of winning, his odds are 4 to 1.

If Hero has, say, an 18% chance of winning, life is more difficult. We have to find a number x such that :

1 / ( 1 + x ) = .18

=>

1 + x = 1 / .18

=>

x = 1 / .18 - 1

In our example, this looks like:

x = 5.56 - 1 or approximately 4.5 to 1 odds against.

Needless to say, this calculation is difficult at the poker table. Fortunately, we don't need to do it all that often. And there are short cuts. But that's the topic for a whole different post.

Just know that odds and percentages are different, and know how to convert between them when needed, offline, and with any calculator, spreadsheet and/or notes you require. And you'll be fine.