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Let's consider a classic hold'em poker game.

We want to calculate the odds of hitting a straight (5 cards in sequence) when we have 4 sequential cards on the flop.

For those who don't know poker, everyone is dealt 2 cards,(followed by the 1st betting round), then 3 common cards are discovered (this is called the "flop", and second betting round), then another card (the "turn", and another betting round), and the last common card (the "river", and the last betting round).

A classical poker table is composed by 9 players, they decide to get involved in the hand by looking at their starting hand. They are more willing to play if their hand has an higher expected value.

The highest 5 cards combination (you can combine yours with the 5 common cards) wins the pot.

So we have 4 sequential cards on the flop and if the turn will complete the sequence of 5 cards we'll have a straight.

What's the probability?

The deck is composed by 52 cards.

Let's consider we have 54, and the flop is 36A. So if a 2 or a 7 comes out we'll have a straight. (8 outs out of 47 undiscovered cards)

$P$ = 8/47 = 0.1702

The odds are 1:4.875.

My question is, do you think this probability vary in dependence of how many players are currently involved in the hand? How to quantify it?

What I mean is, since players get involved in a hand when they have good cards, in an hand where there are fewer players we can suppose the other players were dealt bad (low) cards, so there are more high cards still in the deck.

So it seems reasonable to me that we are more likely to hit a low cards straight in a hand with more players (they probably have high cards) then in a game where there are fewer players (they probably have high cards, but many low cards have been discarded pre-flop by the others).

What do you think? How can I quantify this in terms of probability and odds?

Best regards. Giorgio

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FYI there's also this SE site: – Bruno Stonek Jan 29 '12 at 16:45
Shorter version of this question: on the river in No Limit Hold'em, does how the players played the previous streets reveal something about their hole cards? Answer: yes. – Louis Jan 29 '12 at 19:58

I've seen detailed analyses on this (sadly no longer online) all of which came to the conclusion that card removal is negligible in Hold'em specifically. This is because too many "bad" hands can contain "good" cards (A3o, K6o, etc), and if pots are expected to be multiway, "good" hands can also contain "bad" cards (33, 78s, etc). I hear it matters a lot more in 5-card draw variants.

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