Probability in carribean stud poker

I know that the total combos are combin(52,5)*47*combin(46,4) = 19933230517200. How can I calculate the number of combinations for the player to win with a royal flush?

Not easily. You have to count the number of combinations that the dealer has a qualifying hand (AK high or better) when the player holds a Royal.

The number of combinations where the dealer qualifies should be about: 4189935. This is simply taken from the wizard's table and dividing by 4 (The number of player royal flushes). This gives a rough (ignoring royal vs royal ties) dealer qualifying percentage of 4189935/[47*combin(46,4)] = 54.63% when the player holds the royal. How to count all 4189935 combinations that the dealer may have via combinatorics is an exercise in futility, imo. I'm pretty sure the wizard used a brute force enumeration of all the possible combinations to analyze the game.

I made my living for a few years using brute force, but I suck at math.

Thank you for the answer, but I need to know how to do this futile exercise. Where is this 4189935 coming from???

Quote: Ladymarion

How can I calculate the number of combinations for the player to win with a royal flush?

The best way would be to write a program and brute-force the calculation. Or you can check here to see the results, which show that 16759740 of the 19933230517200 combinations result in the player winning with a royal flush (this figure excludes situations where the dealer does not qualify, and situations where the dealer qualifies but also has a royal flush).

Quote: Ladymarion

Thank you for the answer, but I need to know how to do this futile exercise.

Then brush up on programming that will count all possible poker hands and rank them. I really need to do this myself to be honest.

Trying to use combinatorics and not using a program to systematically count all combinations, () = dealer up card, will just open the door wide open to math errors:

Player...Dealer
AsKsQsJsTs...7s5d4h3c(2s) = dealer doesnt qualify
AsKsQsJsTs...7s5d4h3c(3s) = player wins with a royal flush
AsKsQsJsTs...7s5d4h3c(4s) = player wins with a royal flush
AsKsQsJsTs...7s5d4h3c(5s) = player wins with a royal flush
AsKsQsJsTs...7s5d4h3c(6s) = player wins with a royal flush
AsKsQsJsTs...7s5d4h3c(8s) = dealer doesn't qualify
AsKsQsJsTs...7s5d4h3c(9s) = dealer doesn't qualify

...and so on for every possible combination.

Doing this for all AsKsQsJsTs will lead to 4189935 occurrences of "player wins with a royal flush", 4189935 is the number of combinations where the dealer actually qualifies. Doing it for all 4 royal flushes will get you the 16759740 value found on the wizard's caribbean stud page under the "player wins with royal flush" row. A properly programmed computer code will do this for you.

Assume you hold a Royal. (There are 4 of them. So you'll have to multiply by 4 if you want the non-conditional probability.)
The dealer has 47 remaining cards to draw from. Comb(47;5) possibilities.
Among these, there are 3 Aces, 3K, 3Q, 3J, 3 10's (let's call these HI) and 4 of each other value (LO).

We need to find how many combinations do not yield a win for the player:

• a Royal for the dealer : 3 possibilities.
• a non-qualifying hand: 695 949 possibilities. (see below)

As a result, a Royal has 695952 chances over C(47;5) to not win: 45.37%
Hence a 54.63% chance of hitting.

Before dealing cards, the chance of getting a winning Royal is 4* ( C(47;5) - 695952) over C(52;5)* C(47;5) : 0.000084%
At 10 000 $ jackpot, you have a meagre 0.84 cent expectation... (But then you have the other winning hands to boost that up, luckily!)


-- CALCULATION -----
To be NQ, a hand must : - show 5 different values; - not with an A/K couple; - not in a straight; - not all of the same suit (flush).
We need to distinguish according to how many HI there are in the hand.

# HI	Values	Suits	=
None:	Comb(8;5) - 4 *	4^5 - 4 **	53 040
One:	Comb(5;1)*Comb(8;4) - 2	3^1 * 4^4 - 3	266 220
Two: ***	[Comb(5;2)-1]*Comb(8;3) - 1	3^2 * 4^3 - 3	288 219
Three:	[Comb(5;3)-3]*Comb(8;2) - 1	3^3 * 4^2 - 3	83 655
Four:	[Comb(5;4)-3]*Comb(8;1) - 1	3^4 * 4^1 - 3	4 815
Five:	0 (this is a straight)	TOTAL	695 949

Notes:
* straights
** subtract the flush possibilities: 4 if no HI, 3 if at least one (because you hold the fourth suit in your hand)
*** from here, take out the A/K hands