Computing hole card to flop improvement

[top-spin]

Forgive me, but I'm not a strong mathemetician. I have a two part question.

1) Turn and River odds
=======================

Let's say you have a pocket pair but you didn't make a set after the flop. What's the probability you will make a set ON the turn.


47 unseen cards,two possible cards to hit, the odds are 2/47 which is 4.25%. If you did not make a set on the turn, then the odds of getting it on the river are 2/46 which is 4.34%, a little better because there are less cards to draw from. I'm hoping I haven't messed up yet.

So, if you have a pocket pair and you don't make a set on the flop, there is a (4.25 + 4.34) 8.59% chance you will make a set by the river.

Is that right?

If that is right, then I can take my number of outs and use the formula above to compute the turn, river or combined odds.

2) Flop Odds
=============

This is where I'm really confused. I've seen a lot of tables in books and online that list flop percentages. That is to say, you hold a pocket pair, this is the probability you will flop a set, two pair, etc.

Not that I care what the odds are of flopping quads, straight flushes, etc. I'm interested in the practical odds, the realistic hands that actually take down most pots, like sets or 2 pairs, etc.

If you hold a pocket pair, the chances you flop a set can be anywhere from 10.7% all the way up to 16% based on who's text you are reading. Most of the info i've seen stays in the 11% range. I'm assuming that for the larger percentages, they're including the fact you might flop quads, but i don't see it making that big of a difference.

What i'm very interested in, is a general formula for computing flop percentages. I'd like to start with my outs, and get a percentage back. For example, flopping a set with pocket pair would be two outs. I'm also thinking that a set might not even use YOUR pocket pair. That's possible too, and then you'd have a full.

In any case, is there a general formula, I'm at a loss to come up with one or research one.

I really appreciate your help!!

[BruceZ]

[ QUOTE ]
Forgive me, but I'm not a strong mathemetician. I have a two part question.

1) Turn and River odds
=======================

Let's say you have a pocket pair but you didn't make a set after the flop. What's the probability you will make a set ON the turn.


47 unseen cards,two possible cards to hit, the odds are 2/47 which is 4.25%. If you did not make a set on the turn, then the odds of getting it on the river are 2/46 which is 4.34%, a little better because there are less cards to draw from. I'm hoping I haven't messed up yet.

So, if you have a pocket pair and you don't make a set on the flop, there is a (4.25 + 4.34) 8.59% chance you will make a set by the river.

Is that right?

[/ QUOTE ]

No.

4.25 + (100-4.25)*4.34 = 8.41%

hit turn + miss turn AND hit river


[ QUOTE ]
If you hold a pocket pair, the chances you flop a set can be anywhere from 10.7% all the way up to 16% based on who's text you are reading. Most of the info i've seen stays in the 11% range. I'm assuming that for the larger percentages, they're including the fact you might flop quads, but i don't see it making that big of a difference.

[/ QUOTE ]

Probability of completing your set = 1 - probability of not completing set, which is 1 - probability of missing on each card, which is:

1 - (48/50)*(47/49)*(46/48) = 11.76%.

This would include quads and full houses with a set as long as you catch a 3rd of your card.


[ QUOTE ]
What i'm very interested in, is a general formula for computing flop percentages. I'd like to start with my outs, and get a percentage back. For example, flopping a set with pocket pair would be two outs. I'm also thinking that a set might not even use YOUR pocket pair. That's possible too, and then you'd have a full.

In any case, is there a general formula, I'm at a loss to come up with one or research one.

[/ QUOTE ]

For n outs, probability of hitting an out on the flop is 1 - probability of not hitting it, which is:

1 - [ (50-n)/50 * (49-n)/49 * (48-n)/47 ]

If you're into combinations, this is the same as 1 - C(50-n,3)/C(50,3). Use COMBIN function in Excel to compute C.

[top-spin]

Bruce, you've really cleared that up! Thanks a lot!

[BruceZ]

[ QUOTE ]
For n outs, probability of hitting an out on the flop is 1 - probability of not hitting it, which is:

1 - [ (50-n)/50 * (49-n)/49 * (48-n)/47 ]

[/ QUOTE ]

typo, should be:

1 - [ (50-n)/50 * (49-n)/49 * (48-n)/48 ]

[Cobra]

In reference to your flop odds question there is no good answer but I do have an apporximation based on number of outs for certain situations.

The equation is 5.6% per out and it is good for the following situations.

AX - probability of hitting at least one ace or more.

Formula 3*5.6=16.8%
Actual 17.3 %

AK and hitting at least one A or K or more.

Formula 6*5.6=33.6%
Actual 32.4%

Having a pair and hitting one or more of your cards.

Formula 2*5.6=11.2%
Actual= 11.8%

[Guruman]

hey look! A legitimate rule of thumb! I LOVE that. /images/graemlins/smile.gif

[top-spin]

Bruce you've really set me straight. I've created a little sheet based on your input, I think it's correct:

http://www.vhot.com/poker/poker.pdf

What I am unable to compute are hands that require more than one cards to improve. For example, I have pocket connectors. It's easy for me to compute flopping 3 to a straight, but not really a helpful statistic. I would like to compute flopping a straight draw, and this sounds a LOT more complicated. same with two suited cards (but maybe easier because we're not looking for a sequence). Again, I can easily compute flopping 3 to a flush, but not very helpful. I guess where i'm falling apart is when a hand needs two or more cards to improve. It doesnt look like a simple out calculation because you have to hit more than once. EG, you have 56, a straight draw means u have to flop 78,47 or 34. If u flop a 7, then u need an 8 or 4. If you flop a 4 you need a 3 or 7. If flop a 3 you can only get a 4 to make the straight draw. Is it beyond the scope to explain it? Should I just basically memorize the percentages? I can't even think about the outs needed, basically 8, but 8 different cards based on what u hit. How do you determine the number of outs to hit in that case? Is it still 8?

Am I on the right track? Same with post flop calculations. If I have 3 to a flush, I know 10 cards will help me but i HAVE to catch two of them. I'm totally unsure of how to calculate this.

Im thinking you HAVE to hit one out of the 10 on the turn, so now you have only 9 cards for the river and you must hit it. Is it something like 1 - (37/47 * 36/46). I know it probably shows my ignorance, but I am aware you have to decrease the number of outs you have when you hit one of them on the turn. I know the actual calculation has to be much more complicated.

I guess in the PDF file you'll see the ? marks for where I'm confused /images/graemlins/smile.gif


Thanks again everyone for your help! And the 5.6% approximation works surpisingly well!

[JonLines]

My first post, woohoo!

Theres a very easy rule of thumb, which you can use for the Turn or river. Its accurate enough for those quick decisions when you only get 3 seconds to decide on the internet!

Basically,

Percentage chance of hitting the turn is (2 * number of outs)
Percentage chance of hitting the river is (2 * number of outs)
And percentage chance of hitting on turn or river is (4 * number of outs)


i.e flush draw, 9 outs, 36% chance of hitting by the river.

As I say, its pretty basic, but when your looking at a flop and you have a flush draw, two over cards and a gut straight draw..... you need something thats quick and easy!

Would be 9 outs for flush, 3 for straight, 6 for over cards.
36% chance on hitting turn, 72% by river. Remember this is only chance of hitting outs, doesn't mean you will win, plus... i think the more outs you have, the less accurate rule of thumb gets, but better than nothing.

[NiceCatch]

Now THAT is useful information. Jon, that might go down as the best ever first-time post! Thanks.

Now what I would find even more useful is the conversion to odds... (i.e. 33% means you need 2:1 pot odds). So... here's the system I just came up with.

Your pot odds, given n outs is (25-n):n

So for the flush draw (9 outs), you need (25-9):9, or 16:9 to call. Remember, this does not take into account implied odds or a bet on the turn! (People sometimes forget that your opponent might actually bet on the turn, thus destroying your pot odds! Whoops...)

So then I guess it would be useful to give a formula for odds to drawing just on the turn. Simply cut your odds in half. So the flush draw odds would become 32:9.

Just realized that the idea of odds might not be entirely clear, so here's what it means. If you have 32:9 pot odds, it means that if the pot is $32, the max bet you should be willing to call if you're willing to chase with those odds is $9. If the pot is a more round number like $100, just scale up; your maximum call should then be about $28.