# Can you settle this? Math question disagreement?

#### Reelsoffun

##### When it comes to gambling, timing is everything.
Hi All,

Me and a mate are having a little argument over a little poker based math event.

It all started with my mate complaining about how many aces are on the "community" board when he has PP etc to which i said on average I would expect to see one about 1 in 2.5ish boards. That then led to that he said it would be different to how many players etc so then it all kicked off, so can any mathematicians confirm who is right with the following.

I state that in poker there is a 5/13 chance of seeing an ace within the 5 community cards, regardless of how many players are dealt in, however my mate is absolutely adamant that it has to be different for if there is 3 players or 9 players on the table. eg more or less chance of being an ace there.

Surely no matter how many players in the hand the community cards have the same chance of having an ace each time? Its still just 5 cards being randomly taken from a shuffled deck each time.

After 2+ hrs of going nowhere of trying to explain that the odds dont change that scenario due to the shuffle ultimately determines where all the cards end up. I can sort of see why hes coming to that conclusion but im 99.999999% convinced im right here and need to know either way before i go insane lol

If im indeed wrong then so be it lol

I would love to hear everyone's thoughts and working out on this.....

#### dionysus

CAG
MM
here's a good probability exercise worth checking>

#### dionysus

CAG
MM
edit - im sure im overthinking but im now in my 'call-a-friend-mode' with my go-to maths expert, bastard lol

Last edited:

#### dionysus

CAG
MM
gods
i had answered pretty simply - now my friend's and I's convo has gone quantum
ill be back

#### dionysus

CAG
MM
before i go insane lol

!
btw, that's where we are now - I can understand the debate

#### nikantw

##### Banned User
Combining poker, statistics and the Monty Hall problem, this is my answer:

If you just take notes of the community cards and nothing else, after many many games you will find that the chance for an Ace is irrelevant to the number of players.

In reality though, when you play, you can see the reactions of the other players and use them to figure out if one of them has an Ace! And that increases or decreases the odds of an Ace appearing in the 5 community cards.

Like in the Monty Hall problem where the key is the behavior of the host, here the key is the behavior of the other players that gives you an advantage.

#### dionysus

CAG
MM
I'll let other minds tackle this.
My friend (one of the most brilliant people I know) and I - like you and yours - started with a simple Q
And we both came up with simple quick answers.
Then again.
And debated for some time.
At some point, our brains began leaking out of our ears once we hit Schrödinger, Hawking territory and decided we needed stiff drinks.

#### gerilege

##### Meister Member
PABrogue
PABnorogue
It is correct that the odds of having an ace on the board is independent from the number of players in the hand. Having at least one ace on the board is no different than having at least one on the top of the deck between the first five cards.
However, the odds of having at least one ace in five cards dealt from a deck of 52 cards is not 5/13. The fact that the first card is not an ace slightly changes the probability that the second card will not be an ace, because now only 51 cards will be remaining containing 4 aces and not 52. The same applies for the second card, only 50 cards will be now remaining and so on.
Therefore it is easier to calculate first the probability for not having at least one ace on the board.
For the first card, it is 48/52, second 47/51, third 46/50, fourth 45/49, fifth 44/48. Since these all need to happen at the same time, the odds is the product (48/52)x(47/51)x(46/50)x(45/49)x(44/48). This is about 65,9%. So having at least an ace (which is the opposite of the previous probability) is 100%-65,9%=34,1%.

#### Reelsoffun

##### When it comes to gambling, timing is everything.
It is correct that the odds of having an ace on the board is independent from the number of players in the hand. Having at least one ace on the board is no different than having at least one on the top of the deck between the first five cards.
However, the odds of having at least one ace in five cards dealt from a deck of 52 cards is not 5/13. The fact that the first card is not an ace slightly changes the probability that the second card will not be an ace, because now only 51 cards will be remaining containing 4 aces and not 52. The same applies for the second card, only 50 cards will be now remaining and so on.
Therefore it is easier to calculate first the probability for not having at least one ace on the board.
For the first card, it is 48/52, second 47/51, third 46/50, fourth 45/49, fifth 44/48. Since these all need to happen at the same time, the odds is the product (48/52)x(47/51)x(46/50)x(45/49)x(44/48). This is about 65,9%. So having at least an ace (which is the opposite of the previous probability) is 100%-65,9%=34,1%.

Hi,
Thanks yes i see my error on the 5/13 now as I wasnt factoring that each card isnt independant. But thanks for the confirmation of the argument that the number of players dont effect the 34.1% chance of at least one ace being present on the community board.

#### gerilege

##### Meister Member
PABrogue
PABnorogue
Hi,
Thanks yes i see my error on the 5/13 now as I wasnt factoring that each card isnt independant. But thanks for the confirmation of the argument that the number of players dont effect the 34.1% chance of at least one ace being present on the community board.

I need to add though that while from a theoretical perspective there is no difference, from a practical poker play perspective there are a couple of influencing factors, since the player's hands are dealt before the flop, turn and river, and preflop action usually gives some indication on the hands out there. So I can also understand why your mate may believe based on practical experience that there is a difference.
Some players will just play any two cards dealt, and if everyone plays like that, there would be also no practical difference. On the other hand, in reality better hands are played more frequently, and when there is an ace in a hand, that is more likely to be played. Therefore the practical probability for an ace on board would be lower than the theoretical probability, especially in a No Limit game when less people see the flop. Similarly, better players will tend to fold hands like A6 in a multi-way pot. So when it is most likely not the case and there are actually a lot of players staying in the hand for example in a Fixed Limit game, I would say that aces are slightly more likely to be still in the deck, so the practical probability will be higher for an ace to be dealt than the theoretical one. (BTW: in most cases pairing the ace will not be enough to win the hand in such a multi-way pot. Biggest multiway pots are frequently won by a mid-high pocket pair that improves into a full house, especially if there are flush or straight opportunities as well).

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