Ask the Wizard! (No. 235)

I'm getting asked about this more and more frequently. Unfortunately, the number of combinations in this game would be an unholy gigantic figure. A brute-force looping program might take thousands of years to finish. However, a good programmer can find short cuts. Weighing the costs and benefits, I don't find this project to be a good use of my time. If I lived in Russia or the Baltics, I would likely feel differently.

For the benefit of other readers, Royal Poker is like Caribbean Stud Poker, with the following added options. It is my understanding that the player may invoke any or all of these options, except he may not invoke options 2 and 3 both.

1. If the player can make two paying hands, which both beat the dealer, and neither hand is entirely within the other, then both are paid. I am not sure whether the ante and/or raise are paid twice. For example, if the player had six cards and could make a straight and a flush, then the player could be paid for both hands.
2. The player may switch one to five cards for the price of the Ante.
3. The player may buy a sixth card for the price of the Ante.
4. The player may buy "insurance" before the dealer turns over his four face-down cards. The insurance bet pays even money if the dealer does not qualify.
5. The player may force the dealer to switch his highest card for the next one in the deck for the price of the Ante wager.
6. There is an "AA Bonus" side bet, which pays 7 to 1 if the player's first five cards are a pair of aces or higher.

I can say that the AA Bonus side bet and Insurance Option should never be taken, and thus are not worth anything. The house edge on the AA Bonus is 12.99%. The following table shows the house edge on insurance to range from 8.57% to 33.57%, depending on the dealer's up card.

Insurance in Russian Poker
Dealer's Up Card	Combinations	Probability	Exp. Value
A	132804	0.335714	-0.328571
K	132804	0.335714	-0.328571
Q	108528	0.457143	-0.085714
J	108732	0.456122	-0.087755
10	108936	0.455102	-0.089796
9	109140	0.454082	-0.091837
8	109140	0.454082	-0.091837
7	109140	0.454082	-0.091837
6	109140	0.454082	-0.091837
5	109140	0.454082	-0.091837
4	108936	0.455102	-0.089796
3	108732	0.456122	-0.087755
2	108528	0.457143	-0.085714

I also know from my page on Oasis Poker that just the option to switch cards lowers the house edge from 5.22% to 1.04%. I tend to think the rule about being double-paid, getting to keep the sixth card (as opposed to switching), and forcing the dealer to switch a card will get the game to a worthwhile player advantage, if you knew the proper strategy. Sorry to pull a Fermat on you, but that is that best I can do at this time.

P.S. I have heard some casinos add a rule that if the player wins, then the ante bet only pushes. This would work significantly in the casino’s favor, I think wiping out any advantage.

For the sake of simplicity, let's ignore the fact that the more tickets you buy the lower the value of each ticket becomes because you compete with yourself. That said, the probability of losing all five tickets is (12/13)⁵ = 67.02%. So the probability of winning at least one prize is 32.98%. There are 7033×13=91,429 total tickets in the drum before you buy any. 91,429-40=91,389 are not big prizes. The probability of not winning any big prizes with five tickets is (91,389/91429)⁵ = 99.78%. So the probability of winning at least one big prize is 0.22%, or 1 in 458.

Yes, absolutely! If your goal is to win or lose $x, if limited to even money games, then you maximize your odds by making just one even money bet. This was once the situation, although not limited to even money bets, on a never-aired episode of “The Casino.” There I was consulted on how to maximize the odds of achieving a win of $4,000, given ordinary games and a starting bankroll of $1,000. I had them bet $100 on the pass line and then $900 on the odds in craps. Unfortunately, we lost. Had we won that bet, I would have had them bet enough to get to the $4,000 goal.

However, if fun enters into the equation, you will get more by making smaller bets for a longer period of time. If you simply want to minimize your expected loss, then don't play at all.

If we ignore the house edge (which is very low in craps if played properly), the probability of winning $500, as opposed to losing $1,000, is 2/3. The probability of 4 out of 5 winning sessions would be 5×(2/3)⁴×(1/3) = 32.9%.