Trying to work out the odds of various mahjong "hands".
There are 136 titles, with 4 copies of each title, and 4 suits, with 4 titles being winds which can not be used in a row.
To win (according to the rules where I am at) :
- A pair and either 4 other sets consisting of either a three-of-a-kind
- Or three titles of the same suit in a row.
I have tried to come up with some numbers for the starting hand, but how to find the probabilities during play?
Best Answer
For the starting hand, you need to define clearly what a winning hand is. You then get to assume that the hand is randomly drawn from all possible hands to get the probability. Later in the game, you need to define exactly what strategy will be followed as well, which can be very complicated. This influences the probability of each hand. It is usually not easy.
For example, let us consider 5 card draw poker. It is easy to calculate the chance of getting each kind of hand on the initial deal. One such is here. But if you want the probability of having (at least) three of a kind after the draw you need to define exactly what cards you will throw away. Presumably if you start with three of a kind you will keep them. You might imagine that you would always keep a pair and draw three. The chance there is $1-\frac{44\cdot 43\cdot 42}{47 \cdot 46 \cdot 45}$, about $18.3\%$. If that is always your strategy, this calculation works. But some people will keep an ace with the pair and draw two, or maybe if it is a small pair and you have four cards of the same suit you will draw for the flush. This is why you need to define the strategy precisely. Once you have done so, the calculation technique is the same: favorable cases/all cases, but the counting can be difficult.