# What's the probability of each poker combination winning the showdown?

I play a poker-like game on a casino site, which offers dynamically changing coefficients for each hand on the table.

Each round (blind, preflop, flop, turn) gamblers can bet that one of each hands or combinations win. For example, I think that this game will be won by the hand number 2, and I put my stakes on that. The casino provides a coefficient for that stake (`3.5` for example), and in case the second hand actually wins, I get my stake times the coefficient of profit.

The same goes to combinations: you put bet that this type of combination will be the winning one. For example, when I bet on Two Pairs winning the game, I bet on ANY Two Pairs winning, but ONLY Two Pairs. If it just so happens that a Straight is on the table, I loose the bet.

So, the interesting part is: you can even bet blind. Which means that no information available: just gambling. BUT. The coefficients are off. Of course, casino takes a margin off of each bet, but some bets are more valuable that others. They offer such coefficients for the combinations:

``````High card wins: 100
Pair wins     : 5.80
Two pairs win : 3.10
Set wins      : 6.80
Straight wins : 5.70
Flush wins    : 8.70
Fullhouse wins: 8.70
Straight flush: 100
Royal flush   : 100
``````

It's interesting to see, that seeing a high card on the river is comparable to seeing a royal flush. Of course, a child would know that you are better off betting on straight flush `100:1` than a royal flush with the same coefficients, but it would be interesting to see real calculations. I've been thinking about it, but nothing comes to mind, except bruteforcing every possible table of every possible 6-hand sets, but `C(52, 12) * C(40, 5)` sounds too brutal.

Any idea?

So, I did bruteforce the thing, but not the whole thing. I took 200000 random samples (a sample contains a table and 6 2-card hands) and found a maximum combination among each of them. Turns out, some coefficients are, in fact, better than others (and some of them marginally), yet none of them are good enough considering the expected value.

``````1709.4017094
6.24024961
3.26679952
7.1857148
6.02119461
9.2042892
9.17010546
116.75423234
645.16129032
nan
``````

The `nan` thing means, that among `200000` samples the program never met any royal flush. So, speaking of the expected values:

``````HC: -0.94149994
OP: -0.06451613
TP: -0.06060606
SE: -0.05555556
ST: -0.05
FL: -0.05434783
FH: -0.05434783
QU: -0.31506849
SF: -0.8450093
RF:  nan
``````

Which means, that betting on One Pair will lose you 6.45% of your money on average. The least money-losing bet turns out to be Straight and anything below One Pair and above Full-House is a dead bet.

• It actually super cool to see, that if everyone at the table sat until the showdown every time, only once in 1709 hands a guy would win with a high card. May 15, 2020 at 16:02