Discounting Pot Odds

[Scully]

I have a theory regarding numbers of outs to drawing hands and the associated probabilities of hitting them in a full ring game. Most (if not all) of the literature I've read suggests that if you hold two spades, for example, and two more spades come on the flop, then you have 9 outs to the flush.

How realistic is it that you have 9 outs? This suggests that the only spades dealt so far are the ones you have and on the flop. Given that in a full ring, 20 of the 52 cards have been dealt pre flop (nearly 40%), surely it would be realistic to work on the basis that 40% of the 13 (i.e.: 5) spades have been dealt pre flop. After taking into account the two on the flop, we should expect there to be 6 left in the deck hence we have 6 outs. This would mean that much better pot odds are required to draw to the flush than if the decision was based on 9 outs.

Similar thinking can be applied to open end/gutshot draws. For the open end draw there are 8 cards to make you a straight, but we can expect 40% (3) of these cards to be in our foes' hole cards, hence we have 5 outs.

I would be interested to see what other people's thoughts are about the theory outlined above - even more so if they use the same one profitably!!!

(Apologies if this has been discussed in an earlier thread - I had a look but couldn't see one).

[the_hawk]

The theory is incorrect. You seem to be misunderstanding how outs are used. On the flop in your flush example, there are 47 cards unknown (to you) of which nine are spades. The chances of hitting a spade on the turn are 9/47. It matters not a jot that some of those 47 cards are in opponents' hands.

So, say you're in a 10-handed game. There are only 29 cards left in the pack after the flop, but your chances of hitting a spade on the turn are still 9/47.

[rayrns]

This has been discussed before and bugsbunny has a numbers way to show that it makes no difference where the cards are. Let me see if I can explain it clearly without his excellent post.

Where you are making your mistake is you are saying there are 3 of the flush cards in your opponents hands. Now instead of 47 unseen and 9 flush cards you are working with 29 unknown cards left in the deck and 6 of them are flush cards.

The difference 47 to 29 is because if you know your opponents have 3 flush cards then you also know that their other 15 cards are not flush cards. 2 cards for each of your 9 opponents.

In each of these cases you are approximately 4:1 to make your flush. I may have made a small mistake in the math but I hope I have made the point clear anyway.

Ray

[seanof30306]

[taz115]

I agree with what has been said above. But you can and should make adjustments to (discount) your outs based on the likely holdings of other players as described in many poker books (particularly the ITH books).

What you are asking is should we assume that on average the players at the table have been dealt an equal distribution of your outs. It actually doesn’t make any difference because the number of outs will be reduced proportionately to the number of unknown cards and your pot odds would be exactly the same. So your idea is already incorporated in the way pot odds are calculated.

It is good that you are questioning the way to derive your chances to improve. It means that you are thinking critically about poker concepts and their application. You'll become a better player for it.

taz115

[mchilger]

There are situations, although rare, where you might discount your outs for the increased possibility that your opponents are holding your outs. This generally occurs in multi-way pots. For example, based on the betting patterns, you might think that an opponent is on a similar draw as yours such as a flush draw. This increases the probability that some of your remaining outs might be sitting in your opponent's hand rather than in the deck. Take an example where you have A8s and it seems like another opponent has a flush draw based on his betting pattern. Instead of nine outs, maybe you discount them to 8 outs based on this possibility (I only discount one since it is very difficult to know with certainty that he has the flush draw, if so, you could discount by two outs).

Of course, headsup, there is always that possibility that your opponent and you hold similar drawing hands. The winner of that hand will be the one who is the aggressor, assuming the draw doesn't hit. In that case, if you are the passive player involved in the hand, meaning you are calling with the intention of folding if you miss, then you might discount a little for the possibility that your opponent holds a similar type of draw.

But in most cases, I generally assume my outs to be randomized between all of the unknown cards in the deck.

Matthew

[the_hawk]

[seanof30306]

[mchilger]

Sean, you're doing exactly what you ought to be doing. If you are drawing to a straight and there is a possible flush draw, you should always discount 2 of your outs. How much you discount them obviously depends on many factors, including how many opponents you are against. Against several opponents, I would disregard the two flush cards and only count 6 outs. If you are on the flop, you need to discount this further, since you could hit on the turn, only to lose on the river.

Discounting outs based on the likelihood that they might not win is completely different than discounting them because you think some of your opponents might have mucked one of the cards you are drawing to.

By the way, I discuss a new way of looking at "outs" in my new Odds book. There is an article coming out in Cardplayer next month which also discusses the same concept. I basically distinguish between an "out" and a "dout", which is simply a discounted out. The new terminology will hopefully remind players that you always need to discount outs for the things you mentioned in your post.

Matthew

[55er]

I dont know if you will find this interesting or helpful at all Matthew but after the first reading of you book the concept I put to use right away and is a good habit of mine now is discounting outs.

This concept really made sense to me and I had never seen it in any book before yours.The games I choose to play in are loose-passive games so discounting outs I think is a huge factor in these kind of games.

The interesting thing about the way you put the section together was that if for example you were on a str8 draw and there were 2 flush cards your discount would work on a sliding scale due to # of opponents.

I now take it even further by looseness of opponents left in the hand.If the opponents will play too many hands and call with weak holdings I will discount less.If they are tight and play well after I will discount more thinking they are on a nut or close to it draw and I will have less outs for certain draws.

Just wanted to say it was a very helpful section for me and it definitly stuck out the most after the first reading.

[mchilger]

Tx 55er. You are not the first to say this and I think that chapter has helped many players. The probability chapter in ITH is one of the more difficult chapters for true beginners, but one of the more important. Due to the complexity and importance, that is one reason why I expanded on it some in Texas Hold'em Odds and Probabilities. Next month there is an article titled Outs and Douts which will be in Cardplayer which you might find interesting if you haven't already read the Odds book.

Thanks for the positive feedback - I never get bored of those!

Matthew