re: Poker & Flush Draw...Calculating Outs
Actually this is pretty interesting and something I'm going to do a bit of work on. I thought of this a long time ago and was told it had no bearing on anything so I lost interest and eventually forgot about the idea.
Traditionally we count all unseen cards as equal. But they are not. There are two types of unseen card. Those we could be dealt and those we couldn't.
So with 9 other players they hold roughly 2/5 of the unseen deck and therefore we should assume they hold 2/5 of the outs.
So in that situation the outs should be discounted 1/3 because to make table maths easier the discount should be a simple number favoring on the optimistic side of rounding so if the table held 0.4 of the deck, you would round down to 1/3 rather than down to 1/2 because you just want to take into account there are less odds but don't want to starve yourself of odds.
So instead of 9 outs at a full ring table, there should be around 6.
Miller, Sklansky and Malmuth talk about discounting odds in Small Stakes hold em but I have never read a book that advocates discounting odds as a function of players sat at the table.
Essentially this means that the smaller the table, the closer your odds become to the true odds. So on a 6 max table the discount should be around 1/5 so the flush is worth about 7 outs
With 4 people the discount becomes about 1/8 and so the flush is worth 8 outs.
With less than that the maths isn't worth the bother as it gets so close 9 its not worth discounting.
So in summary
full ring discount outs by 1/3
6 max discount outs by 1/5
4 max discount outs by 1/8
heads up - No discount needed.
Always round the discounted figure up and not down.
This would apply to all outs and not simply flush draws.
I see that this goes against traditional logic, but It does fit in nicely with the concept that play loosens up as the table size reduces.. Is this partly because draws increase in strength? Not all individual draws are as effected as greatly due to the rounding up. But the total outs for a hand should first be calculated and then the total outs be discounted in this way.
I also hear a lot of 6 max players say the enjoy 6 max more because they get more action, their bets are more respected and they dont get sucked out as often.. Is this because they have more outs for the same draw than a 9 handed table?
Makes for an interesting discussion.
Feel free to look over the maths and correct it if necessary but the actual discount figures are incidental, at this point, to the arguments for and against whether the discount is needed.
Same story for two card draws: the other 9 players don't matter because even if they hold a few outs, they also hold a bunch of cards that *aren't* your outs.
For completeness, in two clicks you can find that the regular two card flush draw (9 outs with 47 unknowns) is calc'd as:
(38/47*37/46)=65.03% of not hitting, or 34.97% of hitting
In your scenario, if you exclude the cards held by the other players (with 5.55 outs on average that remain available), then that leaves 29.55 unknown cards so hopefully that gives the same EXACT answer:
(24/29.55)*(23/28.55)=65.03% of not hitting, or 34.97% of hitting.
Now I do like that arugument too.. can we get others who have a good knowlege of statistics to compare the two ideas? One is a case for the discount and the other against it. And discuss which is correct and why.