Counting Outs – Taking into account impact of Hole Cards?

[MJU]

I am new to taking poker seriously and have been working my way through SSHE, TOP and HEPFAP.

I posted a question in the beginners section and whilst I got an answer it didn’t fully answer my question. I’ve since thought about my question some more and think I have found the solution but wanted to check it on this forum.

If we take the example of 4 to a flush with 1 card to come the guidance is I have 9 outs from the 46 unseen cards, therefore my draw has odds of 5.1/1 (46/9). If the pot is greater than 5.1 times the cost of the bet to see the river card then it is +EV for me to see the river.

Now where I was confusing myself was thinking, hell, I don’t have 9 outs as I know on the basis of probability that some of those 9 outs will have already been taken by the hole cards of the other players at the table. Lets say the table is 10 handed then with 1 card to come it would actually be split 3.52 outs already taken in the 18 unseen hole cards and 5.38 remaing to be hit in the 28 cards still in the dealers hand.

This was the math that confused me at first but it is now clear to me that my odds of 5.1 stay the same as I now have 5.38 cards to hit out of 28 available. May be obvious to some but it took me a while to work this out.

Overall, this way of thinking has no impact on the odds and therefore decision making, however, from a psychological perspective it feels worse only having 5.38 outs on average as opposed to 9. I guess the same principle applies to open ended straight draws etc but ultimately the odds of hitting your card are going to remain the same.

Having had these thoughts and worked it through I certainly feel better informed knowing that there are only 5.38 cards out there on average that help me and not 9, even though the odds remain the same.

I am now happy to use the normal guidance going forward, ie 9 outs to a flush but I was wondering if there are any more complex situations where discounting other players hole cards has benefits; overall I can only see it making the situation much more complex than it needs to be.

Also, is the reasoning behind why hole cards are not discounted discussed in any 2+2 books or forums posts – I haven’t come across it in any of the books mentioned above and struggle to use the forum search function effectively.

Many thanks

[kowboy]

Its closer to 4 to 1 on improving not 5 to 1. With a 52 card deck theres 2 in your hand plus 3 on the flop therefore theres 47 cards not seen by you. You cant count discarded hands because they are unseen to you. Good luck.

[BruceZ]

This thread (http://archiveserver.twoplustwo.com/showflat.php?Cat=&Board=probability&Number=566041& Forum=f11) has my answer to a similar question.

[binions]

[ QUOTE ]
If we take the example of 4 to a flush with 1 card to come the guidance is I have 9 outs from the 46 unseen cards, therefore my draw has odds of 5.1/1 (46/9). If the pot is greater than 5.1 times the cost of the bet to see the river card then it is +EV for me to see the river.


[/ QUOTE ]

Stop. There is a difference between odds and chances. Your CHANCE of hitting hitting your flush on the river is 9 out of 46. Your ODDS are 37:9 against (4.1:1) - 37 cards don't help you vs. 9 that do.

It is better to think in terms of ODDS instead of chances when deciding to call a bet based on what's in the pot. For example, if there is $4.50 in the pot, and it is $1 to call, the pot is offering you 4.5:1 odds. If you have a 4.1:1 shot to hit, you call because the pot is offering better odds (an "overlay").


[ QUOTE ]

Now where I was confusing myself was thinking, hell, I don’t have 9 outs as I know on the basis of probability that some of those 9 outs will have already been taken by the hole cards of the other players at the table. Lets say the table is 10 handed then with 1 card to come it would actually be split 3.52 outs already taken in the 18 unseen hole cards and 5.38 remaing to be hit in the 28 cards still in the dealers hand.

This was the math that confused me at first but it is now clear to me that my odds of 5.1 stay the same as I now have 5.38 cards to hit out of 28 available. May be obvious to some but it took me a while to work this out.

I am now happy to use the normal guidance going forward, ie 9 outs to a flush but I was wondering if there are any more complex situations where discounting other players hole cards has benefits; overall I can only see it making the situation much more complex than it needs to be.

Also, is the reasoning behind why hole cards are not discounted discussed in any 2+2 books or forums posts – I haven’t come across it in any of the books mentioned above and struggle to use the forum search function effectively.

Many thanks

[/ QUOTE ]

The reason outs are not discounted in the way you describe is because the ratio stays the same, and the discounting process is superfluous and wastes time.

Unless you have seen an exposed card, or have a solid read that one or more of your outs are blocked, it's simplest and best just to judge your outs and odds based on 47 unseen cards on the flop, and 46 unseen cards on the turn.

If you have a solid read, then you can adjust accordingly.

Example: 3 hearts on the flop. You have AhKd. Someone starts betting like crazy on the flop. You know that small flushes try to protect their hand by betting out for fear of another heart falling, so you put him on a small flush.

If your read is correct, then you are drawing at 7 outs out of 45 unseen cards on the flop (and 7 of 44 on the turn).

[MJU]

Excellent post Binions.

Thanks for taking the time to walk through an analysis of my thinking.

[maurile]

[ QUOTE ]
Now where I was confusing myself was thinking, hell, I don’t have 9 outs as I know on the basis of probability that some of those 9 outs will have already been taken by the hole cards of the other players at the table. Lets say the table is 10 handed then with 1 card to come it would actually be split 3.52 outs already taken in the 18 unseen hole cards and 5.38 remaing to be hit in the 28 cards still in the dealers hand.

This was the math that confused me at first but it is now clear to me that my odds of 5.1 stay the same as I now have 5.38 cards to hit out of 28 available. May be obvious to some but it took me a while to work this out.

[/ QUOTE ]
Yes. You could go further and say that, just as 3.52 of your outs are in your opponents' hole cards, another 2.34 of your outs are stuck in the bottom 12 cards of the deck and thus cannot appear on the river.

So instead of having 9 outs out of 46 cards, or 5.48 outs out of 28 cards, you really have 3.13 outs out of 16 cards.

Oh, but those two burn cards have already been blown, and there's going to be one more burn card as well. There go another 0.59 of your outs.

So you've really got 2.54 outs out of 13 remaining cards.

Or maybe you've got 0.196 outs out of one remaining card.

It's all the same. But 9/46 is the easiest way to do it.