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Ask the Wizard #170

I started to play $5 single-line Jacks or Better recently. Since I started the number of times I have had four to a royal after the draw is 170, while my number of royals is zero. What are the odds of this?

Steve from Oxnard

In 9/6 Jacks or Better with perfect strategy you will see a royal on the draw once every 40,601 hands, but four to a royal once every 460 hands. For every royal you see, you will be one card away 88.33 times. Of the four to a royal hands, 50.37% will pay nothing, 24.89% will pay as a pair, 7.89% as a straight, 16.16% as a flush, and 0.69% as a straight flush. Here are the exact numbers.

Possible Outcomes in 9/6 Jacks or Better

Hand Combinations Probability
Four to royal + straight flush 299529168 0.000015
Four to royal + flush 7005972000 0.000351
Four to royal + straight 3420857076 0.000172
Four to royal + pair 10793270244 0.000541
Four to royal (non-paying) 21844510692 0.001096
Royal flush 490952388 0.000025
All other 19889375425632 0.9978
Total 19933230517200 1

The expected number of royals for 170 four to a royals is 170/88.33 = 1.92. The probability of seeing zero with a mean of 1.92 is e-1.92 = 14.59%.

You mentioned in one of your articles an upcoming appearance on "The Casino" (apparently, it’s been cancelled). I have searched and searched to no avail in finding some kind of link to his episode. I find the idea of a story involving his advice to some young gamblers and how to most likely turn $1,000 into $5,000 quite intriguing. Please respond with some insight/leads as to how I might go about finding a copy of this episode online or purchase a video recording of it, or at the very least come across a written transcript of the episode. Thank you for your time.


Yes, there was a story taped in which some frat boys at UNLV were trying to parlay $1,000 into $5,000 to buy a high end television. They sought out my advice on how to best achieve this goal quickly. I was limited to the games at the Golden Nugget. The Nugget has 10x odds in craps, which I felt offered the opportunity to achieve the goal. It was my strategy on each come out roll to bet min(bankroll/11, (5000-bankroll)/21), subject to convenient rounding, and take the maximum odds. This way we would never go over $5,000 after a 4 or 10 win, would always have enough to take full odds, and would risk the maximum amount if we didn’t have enough to get to $5,000.

For the first bet, this formula would call for a pass line bet of $90.91, but I rounded it up to $100. Then a point was rolled, I think a 6 or 8. On the second roll the shooter sevened out. So the entire grand was lost in two rolls. It apparently didn’t make for very entertaining television and that story never made the air.

Two questions I can anticipate being asked would be (1) why did I have them bet the pass as opposed to the don’t pass, and (2) why didn’t I bet $91 on the line and $910 on the odds, adding the extra dollar out of my own pocket. To answer the first question, I think that for purposes of going for a quick big win the pass line is better. While the overall house edge is less on the don’t pass, I felt it would have taken more rolls to achieve the $5,000 goal, thus exposing more money to the house edge. To answer the second question, there is not much difference between 9x odds and 10x odds and I thought it would look better on television to be betting only black chips, at least to start.

I have used your site to knock down myths and betting systems with many friends and your proofs always win the day. On the Pink Panther (and some others) there is a bonus round that presents a screen with pictures from which to pick. Behind some are coin amounts and others have a symbol that ends the bonus round. Once you are presented with the bonus board the placement of the symbols cannot be changed, can they?

Jack from Rockaway, NJ

Thanks for helping in the fight against betting systems. First let me say that I have never worked for a major slot machine company and don’t have direct knowledge of this. However, I know many people in the industry and those I trust pretty much are in agreement on this topic.

That said, it is my understanding that in all forms of electronic games, including video slots, video poker, and video keno, the outcome is usually determined the moment you make your decision. Meanwhile the possible outcomes are constantly being shuffled, thousands of times a second. I can’t speak for every slot machine but I believe that with the major U.S. slot makers the outcome is not predestined but depends on the exact microsecond you press the button to make your play.

Last night I was playing a friendly hand of poker with relatives and I was more or less trying to get a response out of my husband when I asked, " Do you have a nine?" When all of a sudden the hostess jumped my ass and said I was soliciting hand advice to which I responded no I wasn’t I was just table talking everyone agreed with her but I think they were just pissed because it was my husband. Is there a rule about soliciting hand advice or directly asking a person what they have? I thought you could say whatever you wanted to at a table unless there were rules about cussing. Please let me know.

Rachelle from Lafayette

I believe it isn’t breaking any rules to ask, but to answer the question certainly would. I’m not making any accusations in your case but in general when a couple plays poker together in a home game, rules against collusion are often broken, causing sore feelings among everybody. The usual infraction is guy advising the gal after he already folded. When I lived in California it got so bad with one couple that when I hosted the game I made a rule that they couldn’t both be in the game room at the same time. So maybe the hostess has had trouble with couples playing poker before and overreacted.

I’m trying to compare the cost of replacing an old refrigerator now in order to save on electrical costs, vs. waiting until it dies to replace it. I can calculate how much cheaper it is to run the new fridge vs. the old one: $37/yr., that’s easy. But how do I factor in the cost of the new fridge? Say the new fridge costs $425. I can’t say that *all* of that $425 is a new expense, because I’ll have to replace the old fridge *someday*, if not now, so I’ll have that new-fridge expense at some point anyway. Let’s say that a typical fridge lasts 14 years and my old fridge is 9 years old, so if I replaced it now I’d be replacing it in 5 years. I tried to make a two-column table, comparing the cost of keeping the current fridge for 9 years and then replacing it, vs. replacing it now, but I didn’t know how to make an apples-to-apples comparison because I didn’t know for how far into the future to consider the costs, and because the fridges are replaced in different years. How do I compare the economics of replacing now vs. replacing later? By the way, this isn’t for my own situation, because my current fridge is probably 30 years old. It’s for, uh, a friend.

Spanky McBluejay from Austin, TX

If you keep the current fridge then in five years you will have spent an extra $37*5 = $185 on electricity compared to a new one. If you replace it now you’ll be out $425 but assuming linear depreciation after five years it will still be worth $425*(9/14) = $273.21. So you will have lost $425*(5/14) = $151.79 due to depreciation. So the cost of depreciation of the new fridge is less than the additional electricity expense of keeping the old one, so I favor buying a new one now.

Not a question, just a sincere thank you. Keep telling people there is no winning system. I retired at age 51 and have averaged $86K per year for the past 11 years playing craps. I definitely want others to not believe they can beat the games. Keep up the good work!

Dennise from Lakewood, CO

I will. Over a long enough period of play 99.9% of system losers will lose and 0.1% will sit there full of self-righteousness, thinking it was skill when it was really just luck.

Great site. Asking this for my own personal knowledge. I was playing a 4 person game of texas hold’em. I was dealt pocket aces. I got the royal flush on the river. I was wondering what the odds are of making the royal flush on the river with aces to start?

Rhythmic from Hoquiam, WA

Thanks. Assuming the royal consists of one of your two aces, the number of ways to make a royal by the river is 2*46=92. This would be the two suits in your pocket aces and the 46 possibilities for the extra card. There are combin(50,5) = 2,118,760 ways to deal 5 cards out of 50. So the probability is 92/2,118,760 = 1 in 23,030. has no zero roulette which they call Super Chance Roulette. Are there any systems that would be effective since there is no zero? Without the zero could one effectively play both black and red at the same time since there is no fear of the zero?

Jon from Danville, New Hampsire

I tried to find that game but the site was down when I checked. However, assuming such a game did exist, the answer is no. No system could be expected to beat it, nor lose to it, over the long-run. The expected value of every system would be exactly zero.