Klopp
Klopp
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March 21st, 2020 at 8:32:03 AM permalink
Four people are going to play poker. To determine the dealer, they are dealt in turn a card until an ace appears. The player receiving the first ace becomes the dealer. Is the procedure fair, that is, does every player have the same chance of becoming the dealer? If not, how much differ the chances of the four players?
Last edited by: Klopp on Mar 21, 2020
bobbartop
bobbartop
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March 21st, 2020 at 8:37:58 AM permalink
Quote: Klopp

Four people are going to play poker. To determine the dealer, they are dealt in turn a card until an ace appears. The player receiving the first ace becomes the dealer. Is the procedure fair, that is, does every player have the same chance of becoming the dealer?



Obviously not.
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DJTeddyBear
DJTeddyBear
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March 21st, 2020 at 11:18:55 AM permalink
Not fair at all.

Most (all?) casinos will deal each player, then use high card, then break ties by suit. Suits are ranked alphabetically. If you're not sure if that means C,D,H,S or S,H,D,C, remember that there's a reason the Ace of Spaces is fancy. Dealers will usually stop dealing if the Ace of Spaces shows up.

As a poker league dealer, I'll say that if the Ace of Spaces shows up before the last person is dealt a card, I usually will deal one more card, only because my automatic motions are faster than my reflexes. Some dealers will continue to deal a card to every player regardless.

Note that for a home game, if you don't want to use suits, you can also brea ties by dealing a second card only to the players that have a matching high card - but you have to decide that before the first deal.


Om a side note, if the winning card is the King of Hearts, I'll often announce the card, but slur it a little and follow it up, as such, "King of Farts. He farts a lot that's why he's King. Or maybe that's why he's trying to kill himself." (He's the suicide king.)
I invented a few casino games. Info: http://www.DaveMillerGaming.com/ 覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧 Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
FleaStiff
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March 21st, 2020 at 1:35:49 PM permalink
Quote: Klopp

Is the procedure fair,

YES. It was used by the ex-fbi profiler turned professional poker player o my favorite tv program so don't say nothin' against it.
DRich
DRich
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March 21st, 2020 at 1:40:27 PM permalink
Quote: FleaStiff

YES. It was used by the ex-fbi profiler turned professional poker player o my favorite tv program so don't say nothin' against it.



Not fair at all as the first person to get a card is more likely than the others to be the dealer. High card or low card it fair.
Order from chaos
michael99000
michael99000
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March 21st, 2020 at 1:52:52 PM permalink
If it were just 2 people ..

First guy has a 7.7% chance of getting an ace

Second guy has a 7.3% chance of first guy not getting one and himself getting one.
DRich
DRich
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March 21st, 2020 at 1:54:42 PM permalink
Quote: michael99000

If it were just 2 people ..

First guy has a 7.7% chance of getting an ace

Second guy has a 7.3% chance of first guy not getting one and himself getting one.



Yes, but if you keep going the first person is most likely to get it.
Order from chaos
EdCollins
EdCollins
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March 21st, 2020 at 9:20:52 PM permalink
Quote: Klopp

If not, how much differ the chances of the four players?

The first player to receive a card will receive the first ace approximately 28.0 % of the time.
The second player to receive a card will receive the first ace approximately 25.9 % of the time.
The third player to receive a card will receive the first ace approximately 24.0 % of the time.
The fourth player to receive a card will receive the first ace approximately 22.1 % of the time.
onenickelmiracle
onenickelmiracle
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March 22nd, 2020 at 3:01:34 AM permalink
What advantage is there in being the dealer? They are proving being the dealer is not fair, but if there is no advantage in being the dealer, isn't it?
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ChesterDog
ChesterDog
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March 22nd, 2020 at 3:46:25 AM permalink
Quote: EdCollins

The first player to receive a card will receive the first ace approximately 28.0 % of the time.
The second player to receive a card will receive the first ace approximately 25.9 % of the time.
The third player to receive a card will receive the first ace approximately 24.0 % of the time.
The fourth player to receive a card will receive the first ace approximately 22.1 % of the time.



An "infinite-deck model" of this procedure gives results similar to yours:
PlayerProb.
10.28077
20.25917
30.23923
40.22083
Sum1.00000


In the infinite-deck model with the constant probability of an ace = 1/13 = a, the players' probabilities of winning are:

P(1) = a / [1 - (1-a)4]

P(2) = a(1-a) / [1 - (1-a)4]

P(3) = a(1-a)2 / [1 - (1-a)4]

P(4) = a(1-a)3 / [1 - (1-a)4]

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