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Ask the Wizard #76

How does one calculate the probability of hitting on a *specific* number of draws in n-play video poker? Example: drawing to a four-card Royal on a triple-play machine, the odds of *at least* one hit are 1-(46/47)3 = 0.0625, correct? But how do you determine the odds of hitting exactly 1, 2 or all 3 Royals?

John from Milwaukee, USA

The probability of hitting x royals in an n-play machine when drawing to a 4-card royal is combin(n,x) * (1/47)x * (46/47)n-x. For an explanation of the combin(n,x) function visit my section on probabilities in poker. In the case of 3-play the probabilities are as follows:

0 royals: 0.937519
1 royal: 0.061143
2 royals: 0.001329
3 royals: 0.000010

Playing 3 Card Poker and using a Shufflemaster, I received 2 consecutive hands with the same cards, same suit (from the same deck). I was sitting at first base, so these were the first cards out both times. What are the odds of this occurring on consecutive hands?

Larry from Silverdale, Washington

There are combin(52,3) = 22100 ways to arrange 3 cards out of 52. So the probability of any given hand matching the last one exactly is 1 in 22100.

Hi Wizard, I’m Andrés Varillas from Lima - Per?, South America, I begun to play just 6 years ago, and I loose 20 thousand dollars in this years, for me is too much, I’m really upset of this situation and I want your help, I offer you if you make me win I receive you here in Per? to take vacations, Per? is a beautiful country, and you enjoy Know it, in the Casino I play, there are many machines of IGT Co., like Catch a Wave, Cleopatra, The Monsters, Leopard, I really loose in 1 of 15 chances, I want to try back my money. Maybe you can give some secrets to win in these machines. I appreciate your help, and send you a big hug.

Andrés from Lima, Peru

Sorry to hear about your misfortune. However that is the fate that awaits most who overplay their bankroll, especially in slots. There is no way to beat any of these games. I would recommend you quit gambling completely.

This past weekend I played Let it Ride at El Dorado in Reno. Side Bet payout was 20000 (Royal), 2000 (Straight Flsh), 200 Four, 75 Full, 50 Flush, 25 Straight, 5 Three, 4 Two pair, 1 High pair. Did not see this covered on House edge what would edge be?

Bayne Steele from Petaluma, California

This pay table has a house edge of 13.07%, the lowest I have heard of for Let it Ride. Still a sucker bet though.

Hello Professor, would it be possible for you to provide an optimal strategy for Crypto’s double bonus poker? Also, could you recommend a strategy generator that will create a near-optimal strategy for any video poker game with any paytable?

Jan from Ontario, Canada

There are two software programs that can produce near optimal strategies for almost any video poker game. One if Video Poker Strategy Master by Tom Ski and the other is Frugal Video Poker by Jean Scott. Winpoker 7.0 promises to offer this feature as well but as of this writing it is not out yet. I don’t like to give away too many video poker strategies for free because other experts have to make a living selling video poker software or strategy cards.

To begin, I am not a mathematician but I am a casino player. I have followed some of your articles in Casino Player magazine and I subscribe to your on-line newsletter. By the way, I hope you had a wonderful time with your family and friends in Seattle.

I just had an eye-opening experience at Casino Windsor. No where do they publish their percent payback on slots. However, that aside, I was going to play quarter (my comfort level) video poker. I was really taken aback when I put up the paytable. They were 5/4 machines. I am talking Jacks or better was only 5 coins on a full house and 4 coins on a flush. I looked at about 20 machines and only found one that paid any better and that was a 6/4 machine.

As I stated, I am not a mathematician but I think that payback percentage must be in the very low 70s. Needless to say I didn’t play video poker there because I know that the longer I would have played the greater the loss I could have expected with the house taking approximately $30.00 from every $100.00 put through the machine. That is not a gamble with some expectation of winning, it is a sure loss for players. On the Detroit side of the river, MGM Grand has their machines at 7/5. Not really great but a whole lot better than 5/4.

Could you please tell me the exact percentage payback on the 5/4 and 7/5 machines. Since none of the area casinos post their average payback on slots I am (and this is dangerous) assuming that their reel slots payback the same percentages. Best regards.

Mort from Walled Lake, Michigan

Actually with perfect play the 5/4 pay table return 92.78%. Still one of the worst pay tables I have ever heard of. Have you tried the Greektown casino in Detroit? I don't know what games they have but I do know they have had security remove several winning video poker players from the building, including a old lady who hit a royal on a machine with a 97% pay table. They must have something good enough to warrant throwing winners out for.

I need to know the odds of someone getting 4 of a kind during a hand of 7 card stud with five players and one deck of cards? I hope you can help me, and Thank You for your time.

Richard from Saint Joseph, USA

There are combin(52,7)=133,784,560 ways to arrange 7 cards out of 52. The number of 7-card sets including a four of a kind is 13*combin(48,3) = 224,848. The 13 is the number of ranks for the 4 of a kind and the combin(48,3) is the number of ways you can choose 3 cards out of the 48 left. So the probability is 224,848/133,784,560 = 0.0017, or 1 in 595.

Michael, saw you on the travel channel from Vegas the other day giving your expert advice regarding house advantage. Good job. Didn’t realize you were that young. I was playing craps at the new Seneca/Niagara casino and instructed the dealer to have the odds working for a don’t come bet on the come out roll. He advised the odds are always working for that bet. Is he correct?

Gordon from Niagara Falls

Thanks for the kind words on my Travel Channel appearance. For those of you who missed it the show was titled Sucker Bets. The dealer was right. In general bets are turned off on a come out roll in which a seven would cause the player to lose. I guess they don’t want some players to sour the mood of the table by rooting against a 7 on the come out roll. Since a 7 would cause the odds on the don’t pass to win they are left on.

I just left a discussion in which we all agreed the Martingale System is not good use. My discussion mates used an unlimited bankroll to make the theory work to their advantage by saying the ONLY thing wrong with the theory is the table limits set by the casinos. I conceded that point, that, yes, the table limits will stop this system. However, I went one step deeper by saying that the system will fail because Martingale assumes that on an unlimited bankroll you will win one time to make the 1 unit profit sought after. I disagree. Albeit the house edge is small on certain types of wagers in a casino, there is no guarantee that the win will ever occur. Granted, losing 1,000 or 1,000,000 times in a row at something is very unlikely, but it is possible. If, like I did, it is assumed that an unlimited bankroll is at our fingertips and there is no table limit - meaning doubling to infinity is possible, can/does the system still fail because the "theory assumed win" is not a guarantee?

Eric from Ephrata, Pennsylvania

I maintain that even with an infinite bankroll, betting limits, and time the Martingale still would not beat a negative expectation game like roulette. My reason is that as these elements approach infinity the expected value of the Martingale on roulette is still -5.26%.

Still, mathematicians I respect have disagreed with me. The debate tends to get very abstract and absurd, hinging on the nature of infinity, which is a man-made construct to begin with. There is nothing known in our universe that is infinite. If forced, I think it is a ridiculous question.