Poker dice is played with five dice each with playing card images – A K Q J 10 9 – on the six faces. There are a total of 6 × 6 × 6 × 6 × 6 = 7776 outcomes from throwing these dice, of which 94% are scoring hands and only 6% are busts.

Here are the number of outcomes for each hand and their probabilities

And here is the data presented as a pie chart

A noticeable feature of the data is that the number of outcomes for 1 pair is exactly twice that for 2 pairs, and likewise the number of outcomes for Full house is exactly twice that for 4 of a kind. But when outcomes are calculated in the conventional way, it is not obvious why this is so.

Taking the first case, the conventional calculation runs as follows:

** 1 pair** – There are

^{6}C

_{1}ways to choose which number will be a pair and

^{5}C

_{2}ways to choose which of five dice will be a pair, then there are

^{5}C

_{3}× 3! ways to choose the remaining three dice

** 2 pairs** – There are

^{6}C

_{2}ways to choose which two numbers will be pairs,

^{5}C

_{2}ways to choose which of five dice will be the first pair and

^{3}C

_{2}ways to choose which of three dice will be the second pair. Then there are

^{4}C

_{1}ways to choose the last die

Conventional calculation gives no obvious indication of why there are twice as many outcomes for a 1-pair hand than a 2-pair hand.

But there is an alternative method of calculation which does make the difference clear.

**– – – –**

**A different approach**

Instead of starting with component parts, consider the hand as a whole and count the number (n) of different faces on view. The number of ways to choose n from six faces is ^{6}C_{n}. Now multiply this by the number of ways of grouping the faces, which is given by n!/s! where s is the number of face groups sharing the same size. The number of face combinations for the hand is ^{6}C_{n} × n!/s!

Since the dice are independent variables, each face combination is subject to permutation taking repetition of faces into account with the permutation-repetition formula 5!/n_{1}! n_{2}! .. where n_{1}, n_{2} etc are the number of repetitions of a given face.

Thus the total number of outcomes for any poker dice hand can be calculated with a single formula:

It is easy to see from the table why there are twice as many outcomes for a 1-pair hand than a 2-pair hand. The number of face combinations (^{6}C_{n} × n!/s!) is the same in both cases but there are twice as many dice permutations for 1 pair. Similarly with Full house and 4 of a kind, the number of face combinations is the same but there are twice as many dice permutations for Full house.

**– – – –**

P Mander April 2018

Hi Jack, this post is actually about poker

dicebut I guess you saw that. I might do more combinatorial analysis on poker and slots, we’ll see. Good luck with Red Dog.LikeLike

I want to say that in addition to the above methods of probability when playing online poker, you can use another one. Just click to read a review at casinobonustips.com of the online poker service that interests you. There you can see the pros and cons, as well as the security level and user trust rating.

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