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Skuzzy
01-17-2005, 09:24 AM
While out for a quiet drink at a local establishment, I got into a discussion with the owner about gambling and cards. He was a brag player. We were discussing the propensity of losing players to play badly at the end of the night and specifically about some of the more unsavoury proposition bets that less than honourable gamblers use to exploit this weakness and take what little money these poor hapless chaps had left.

He became suddenly silent and then went a little pale before explaining to me the game of shoot that he and his friends had often played in his youth.

One player (always the same friend from his experience) puts up a pot of say £50. Each player is then dealt three cards with the exception of the dealer. There's a fixed minimum (ante) of 50p. A player can increase the size of his bet on seeing his hand and the dealer has to match the bet up to a maximum of what he has in the pot.

The dealer then turns the top card. The player wins if he has a card of higher value in that suit in his hand. If he has no card of that suit he loses.

The dealer then proceeds onto the next player and repeats the process.

Just thinking in terms of outs for the dealer I came to the conclusion there was a huge overlay in his favour. I made a few quick calculations:

22100 hands

Dealers chances against big hands:

AAA 13/49 26.53% 36:13 2.77

KKK 16/49 32.65% 33:16 2.06

QQQ 19/49 38.78% 30:19 1.58

JJJ 22/49 44.90% 27:22 1.23

TTT 25/49 51.02% 24:25 0.96

AA and any suited card 26/49 53.06% 23:26 0.88
AA3 rainbow 24/49 48.98% 25:24 1.04

Anyway, just curious to know if anyone has seen this played and if anyone can help me calculate the total number of winning versus losing hands. My friend is very keen to know just how badly he was being swindled.

pzhon
01-17-2005, 11:51 AM
The players may be able to get some information from each others bets, but for simplicity, assume that there is only one player. I see no reason to bet anything between 50p and 50 pounds. Most of the time, you just bet the ante, and lose a few pence. When you have 25 or more cards covered, you raise to 50 pounds.

Let's separate the game into two pieces. In one, the player doesn't play unless it is favorable. If the hand is good, the player bets 99 antes. In the second, the player bets 1 ante no matter what.

/images/graemlins/spade.gifOnly 1456 of the 22100 hands are favorable, 1/15.18, but on average, those are worth 13.13 times the ante after you bet 99 antes antes, for an average gain of 0.8652 antes.

/images/graemlins/spade.gif Since the player has no choices, let the dealer pick a card first, then deal the player's hand. If the dealer draws an ace, the player loses. If the dealer draws a king, the player has 1 out 3 times. If the dealer draws a deuce, the player has 12 outs 3 times. The player wins with probability .3038, so this game is worth -0.3924 antes.

Put together, the original game is worth 0.8652 - 0.3924 = 0.4728, so it is definitely in the players' favor, even without the ability to gain information. It would still be good for the players if they were only able to raise to 50 times the ante. Are you sure the ante was 50p, and not 1 pound?

Of course, in practice, I would expect people to raise to less than the full amount when they are only a 25:24 favorite, and these misplays help the dealer. If players raise losing hands, that would be quite costly. If only one player can raise to the maximum even if several players have good hands, that helps the dealer, but not by much since good hands are so rare.

Skuzzy
01-17-2005, 01:32 PM
The pot size was just a number I picked. I asked and he said he reckons £10 was the pot size to start. Additionally, if a player 'shoots' for the pot he actually takes control of the pot and becomes the dealer. This wasn't clear originally.

Apparently the players in general played scared. His recollection is that he himself only ever played for the whole pot a couple of times. The 'regular' dealer however is remembered as frequently, by comparison, playing for the whole pot if he lost control, and nearly always walking away with several of the players wage packets. It seems he had an unbeatable table image, understood the game and manipulated it to his advantage.

Thanks for your thoughts. Nicely done. How did you get 1456 favourable hands though? I was having trouble seeing a simple way to calculate this 'probably' because my grasp on probability is weak.

pzhon
01-17-2005, 10:59 PM
[ QUOTE ]

Thanks for your thoughts. Nicely done. How did you get 1456 favourable hands though? I was having trouble seeing a simple way to calculate this 'probably' because my grasp on probability is weak.

[/ QUOTE ]
You are welcome.

This falls under combinatorics, not probability. All favorable hands have 3 different suits. Fix the suits. To separate the winning hands among the 13^3 possibilities, I used a generating function for the number of cards covered, since that has a simple form:

(1+x+x^2+...+x^12)^3 =

1+3x+6x^2+...+78x^25+66x^26+55x^27+45x^28+36x^29+2 8x^30+ 21x^31+15x^32+10x^33+6x^34+3x^35+x^36

The coefficient of x^35 is 3. That means there are 3 hands that cover 35 cards: AAK, AKA, and KAA. The winning hands cover 25 or more cards, so the number of winning hands is the sum of the coefficients of powers of x from x^25 up, 78+66+55+45+...+3+1 = 364. (I just added these using Mathematica by evaluating the top terms at x=1, but it isn't just a coincidence that 364 = 14 choose 3.) After multiplying by the 4 choices for the missing suit, we get 4*364 = 1456 winning hands.