Probability: Difference between revisions

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(Probability: WIP. Don't shoot the messenger.)
 
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**WORK IN PROGESS**
'''Probability''' in mahjong is a concept that can be applied to many different situations. Because unlike chess, go or shogi, mahjong is an incomplete-information game.  Players are required to predict outcomes and chances of reaching [[tenpai]], improving their [[Shanten|hand state]] (lowering tenpai or raising [[uke-ire]] (the useful tile count)), and completing their hand.  At the same time, players try to project and [[Defense|guess opponent waiting tiles]]. Naturally, some of these are difficult concepts to describe with mathematical models, due to various [[Situational mahjong|game situations]].
 
'''Probability''' in mahjong is a concept that can be applied to many different situations. Because unlike chess, go or shogi, playing mahjong is an incomplete-information game, requiring players to predict outcomes and chances of completing their hand, reaching tenpai, improving their hand state (lowering tenpai or raising uke-ire (the useful tile count)), as well as attempting to project what an opponent can be waiting on. Naturally, some of these are difficult concepts to describe with mathematical models.


== Basic concepts and variables ==
== Basic concepts and variables ==
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=== Chance of drawing from a subset (tsumo chance) ===
=== Chance of drawing from a subset (tsumo chance) ===
S = 1 - notS; notS = ({Sum: (tile=0 to 33)} numCopies(tile) / remTiles)^remDraws<sub>E</sub>
S = 1 - notS; notS = ({Sum: (tile=0 to 33)} numCopies(tile) / remTiles)<sup>remDraws<sub>E</sub></sup>


=== Chance of calling a final tile from a subset (ron chance) ===
=== Chance of calling a final tile from a subset (ron chance) ===
R = 1 - notR; notR = ({Sum: (tile=0 to 33)} numCopies(tile) / remTiles)^(remDraws<sub>E</sub> - remDraws<sub>X</sub>)
R = 1 - notR; notR = ({Sum: (tile=0 to 33)} numCopies(tile) / remTiles)<sup>(remDraws<sub>E</sub> - remDraws<sub>X</sub>)</sup>


=== Chance of winning (raw chance, at tenpai stage) ===
=== Chance of winning (raw chance, at tenpai stage) ===
W = 1 - (notS * notR); note that raw chance will give numbers whose total can easily exceed a total probability of 1. This is to be expected.
W = 1 - (notS * notR); note that raw chance will give numbers whose 4-player total can easily exceed a total probability of 1. This is to be expected.


=== Chance of not having a win declared in one full turn ===
=== Chance of not having a win declared in one full turn ===
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