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What is exactly implied odds, and how to calculate it correctly, in which situations should someone use this tool, and in which situations should someone avoid it?

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To understand Implied Odds (IO) it's useful to clarify what It's counterpart is, Explicit Odds (EO). 

EO describes how much you will win immediately in relation to what you have to risk. This is described in terms of a ratio, Total pot size : Amount we have to call. For example, current pot size is $50. Your opponent bets $50. Therefore, the current pot size is $100, and you must call $50. EO = $100 : $50 = 2 : 1

IO describes how much you will win on future streets if you hit the hand you are drawing to. It's calculated similarly to EO except you add what your opponent has left, not including his current bet, to the current pot size. Total pot size (+opponents remaining stack) : Amount we have to call. For example, current pot size is $50. Your opponent bets $50. Therefore, the current pot size is $100, and you must call $50. Opponent has $50 left.  IO = $100 (+$50) : $50 = 3 : 1

Importantly, this particular IO calculation assumes that you will win the entirety of what your opponent has left. This obviously won't always be the case. That's where your judgement comes in. "If I call now, how much more can I win from villain on future streets?". The part of the calculation regarding the opponents remaining stack is subjective. 

It's worth noting that there exists a third paradigm of pot odds, and that is Reverse Implied Odds (RIO). These odds describe how much you will lose on future streets if you hit the hand you are drawing to. If you're already the favorite in the hand, then this applies to your opponent. It's wise to consider the possibility that the hand you're drawing to may not be best, even when you make it. 

Using these tools will help you decide whether to continue in a hand or not. Which is to say you should be using them every hand you ever play.

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In limit poker, implied odds refer to what you can collect in future bets from an opponent, if you complete your draw (e.g. to a straight or flush) and make your hand.

Let's say that you are playing $5-10, and your opponent has bet $10 on the turn with say, a pair, making a pot of $90. If you are drawing to a straight or flush (4- to -1 against), those are long odds. It looks like you shouldn't call for $10.

But if you make your straight or flush, you will be able to trap your opponent for another $10, for a total of $20. Given that fact, it becomes a much closer issue. Most people would call under those circumstances (even though they shouldn't, if there is the possibility that the same card that makes your straight or flush makes the opponent a full house.

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