3

As far as I understand, odds of hitting a draw are calculated like this:

On flop-to-turn:

(47/outs) - 1 = result-to-1

On turn-to-river:

(46/outs) - 1 = result-to-1

The above are calculations of chances to hit a draw when next card comes, right? Then how to calculate the odds of hitting a draw in flop-to-river?

6

I'm not entirely sure what you are trying to say with the math that you have in your question, but I think you are trying to show how you get the odds of hitting a flush or a straight on the turn or river when you have 4-to-a-flush/straight-draw on the flop. The same basic strategy of calculating odds can be done to see what your chances are to hit a set on the flop with you have a pocket pair, or the odds to "fill up" a full house.

I believe this has been answered previously in this post What are the odds I will hit my flush?

However, to answer your question, calculating flop-to-river - you basically add the flop-to-turn and the turn-to-river odds together. This is because you have two chances to hit your card, thus the addition. The real odds include handling the case where you don't hit your card on the turn (which I include in the math).

The equation is: (call out to @configurator for the math from the previously answered question)

(outs/47) + ((outs/46)*(1-(outs/47)))

The term (1-(outs/47)) indicated the chance that you didn't hit on the turn. Because if you did, then you wouldn't need the river card, would you.

So, for a flush - where you have 9 outs with a 4 flush on the flop - the odds are:

0.1914 + (0.1957 * (1-0.1914)) = 34.96% (about 1 in 3).

And the odds to complete an open-ended or double-belly-buster straight are:

0.1702 + (0.1739 * (1-0.1702)) = 31.45% (about 1 in 3).

However, when they are dependent events, where you need two running cards (like a runner-runner flush on the turn and river) the odds are multiplied

So, for a runner-runner flush (not that this was asked but I'm gonna put it in anyway) - where you have 3 to a flush on the flop - the odds of completing the flush are:

(10/47) * (9/46) = 0.2128 * 0.1957 = 4.07%.

Slightly better than a 1-outer on the river - which is 2.17%.

So, the next time that donkey beats you with that runner-runner suck out, you can tell them exact what their odds were - and watch them not care as they rack the chips. Enjoy.

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  • +1 for correctly communicating when you have to ADD the odds vs. when to MULTIPLY them ;) – Radu Murzea Sep 23 '14 at 9:25
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I just have to add another answer to this question. I added it as another answer as opposed to a comment. I hope that's good etiquette, if not I apologize.

The previous answer is "correct" in the purely mathematical and pedantic sense. There are lots of answers like the above so I'm not sure I'm adding anything. But, that is not how I actually calculate odds when I'm actually playing poker.

Firstly, I'm not sure it's all that important to have exact percentages. It's important to know what class of odds you are in (1 in 3, 1 in 5, 1 in 8, Coin Flip-ish, etc.). To that end, you can have much of these odds calculations canned and ready to go as you see the situation unfold. See: TEXAS HOLD'EM POKER ODDS & PROBABILITIES. So you certainly don't need to calculate odds on the fly. If you flop 4-to-a-flush, then you have a 34.96% chance of hitting your flush on the turn or river. If you miss on the turn, then you have a 19.6% chance (1 in 5) of hitting your flush on the river (not that it would win necessarily, but you'd have a flush).

Or, conversely, if you flop something like top pair and you put your opponent on a flush/straight draw, then it means you have a 65.04% chance to win, etc.

All that means is that you should make your decision to go forward on a draw or make your decision regarding the amount to bet based on that percentage - make sure the player drawing is not getting the pot odds to call.

However, when I need to actually calculate odds - like when I need to calculate how much to bet or how much the pot needs to be to call a bet - I do a very simple calculation.

Each cards is worth about 2 percent. 1/52 is 1.9% but that's a awkward number. 2 is nice. My "quick calc" is: count up my "outs", multiply by 2 and multiply by the number of chances to hit.

Outs * 2 * "cards to come" = odds to hit

The result is the odds I work with while actually in a hand. So my "quick calc" for a flush draw is: "9 outs" * "2% per card" twice (because you add the turn and river) which is 18 + 18 or 36%. It gives me the class of odds. The real odds are 34.96%. My quick calc is off by 1.04%. If 1.04% makes a difference in your decision then rock on. Not so much for me.

So, if I have a pocket pair - like 44 - and I want to call a raise that I read as AK or some big pair, I need to see the right odds in the pot or be reasonably sure that my implied odds are in the right class. The "quick calc" odds for flopping a set - which is really what you are looking for - is "2 outs" * "2% per card" three times (for each card in the flop). That is 12%. That is about 1 in 8. Not great odds, but if you have 2 other aggressive players in the hand you can clean up. You all know what I mean - we live for that flop. But it's good to know the odds.

The real odds of flopping a set are: 2/50 + (2/49*(1-2/50)) + (2/48*(1-(2/49)) = 0.04 + 0.0392 + 0.0399 = 11.9%

I think my "quick calc" works just fine.

If I ever find myself "multiplying" odds because I need a runner-runner, or something - which is (Outs * 2)^"cards to come" - I just fold - or go for a bluff. ;-)

When I think I'm ahead in the hand, but someone is trying to run me down, I use the same calculations. I count up their outs based on my read, and figure out how much I need to bet to give my opponents bad odds. It doesn't bother me if they actually catch the card and make their hand - hey, it happens - actually, 34.96% of the time on a flush draw. However, I know they made a bad call. If I can get them to do that often enough, I will eventually win. It's a mathematical certainty.

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Probability of a runner-runner flush is: (10/47) * (9/46) = 90/2,162 = .041628

That's just under 4.2%. About 1 chance in 24 or 23 to 1 against.

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from practical point of view, easiest way is to put the situation in a poker odds calculator, like this one.

Another approximation which is the rule of 2 and 4. each card have a chance of 2% to show up on each street (because 1 out of 50 is 2% and there are about 50 cards in a deck). In this example we have 9 outs, since we have 2 streets that's 9*4=36%.

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