Lets say I am holding 10s9s.

The board is 4s5s7d

Its 9 handed (including me). 7 of the people have folded, and im heads up against an opponent (not sure this matters though).

Is it safe to assume of the 16 other cards dealt, that at least one of them is one of the remaining 9 spades in the deck?

Do I also include the 3 burn cards that will be burnt, in the odds if the hand goes to completion, like for example if I go all in on the flop and get called.

Or do I just factor in one burn card, the one used on the flop.

Or do I factor in no burn cards. I have tried almost all combinations and can't seem to figure out a percentage chance that makes sense to determine this answer.

So far my speculation is that its 9 (representing the remaining spades in the deck) divided by 19 (the 16 other cards dealt + the 3 burn cards)

which 9/19=.4736842105263158

or 47.4% chance roughly, therefore concluding that its NOT safe to assume this because its less than a 50% chance, and we need a more than 50% chance for it to be a profitable assumption when calculating the drawing odds.

I ran 10 samples in real life dealing out cards (which I know, is way too small a sample size most likely to account for deviations) after properly shuffling each time, and there was one of my suit every single time in the sample, and often more than 1, and a few times 3 or 4 of them.

How do I calculate this correctly? because my next question is the same question about open enders, or 8 outers, vs. 2flush draws (9 outers). Is it safe to assume one of the 8 outs will be dead between the burn cards and the other hands dealt?

A thorough explanation and answer is so much appreciated. I do not necessarily think I am correct at all in my calculations which is why I am asking. Can whoever answers also explain the formula you would use to solve this and why?


EDIT: I read the stickied post about flush draws AFTER I posted this. I play in a live casino, so there will certainly be burn cards. And to be honest, I'm not quite sure I understood the answer in the stickied post or if it is the same example. (I got lost on the C(9,0) notation- it has been more than a decade since I have taken statistics). So I'm trying to understand in layman's terms what computation is relevant and why. I need it dumbed down so to speak.

If you start subtracting outs because 'they are missing from the deck because they were already dealt to somebody" Then you have to start subtracting NON-outs because they were ALSO dealt out to people.

Use the simple formulas and they will treat you well....

4x (clean outs) after the flop = 'hit%' by river
2x (clean outs) after the turn = 'hit%' by river

If you have more than 8 outs on the flop, you can get a more accurate percentage by subtracting 1% for each out over 8.

For Example, if you have 15 outs after the flop 15x4=60% -7=53%

Without showing any math, here is a simple explanation offered by David Lyons showing that unknown opponent holdings should not be a factor is estimating board probabilities.

“…you deal the hands out and get two hearts in your hand. Then let’s say one of the other players just simply swaps his hand with the bottom two cards on the deck. Would that affect the probabilities of hearts on the flop? Obviously not. So why do you need to consider what's in the other players hand at all? “

Of course, if you make an estimate of an opponent’s hand based on his actions and the dealt board, then you have to account for that.

It's safe to assume that 16*9/47 of them are spades, that's the average. If you also factor the two burn cards, make it 18*9/47. Then the probability of the turn being a spade is (9 - 3.4468)/(47-18) = ...wait for it... 9/47, the same as not assuming anything about the unknown cards. Nor will the bottom 26 cards in the deck affect anything. The bottom line is that each unseen card has the same 9/47 chance of being a spade, so it doesn't matter which of them is dealt as the turn. The presence of folded hands just means you get a card 17 cards lower in the deck, but that card has the same chance as any folded card (or what would have been the top card had nobody folded).

You can certainly make an estimate on what others have folded based on how they play. But, except in rare cases, no one is that good to do it with any reliability.

I once spent way too much effort doing the folded outs & folded non-outs calculation. Unsurprisingly the result was the exact same equity for my hand anyway, with extra steps.