Ask the Slot Expert: Do royal flushes really happen every 40,000 hands?

Question: I am a regular reader of your column and want to express my appreciation for you and its contents.

I am a regular player of 88 Fortunes and here are three reasons why:

1. September 2015 - Hit the Grand for $12,000+ playing $1.76 at the Eureka Casino in Mesquite Nevada. (They have two machines.)
2. March 5, 2016 - Hit the Grand for $18,77.81 playing $1.76 at CasaBlanca Casino in Mesquite Nevada.(They have four machines.)
3. September 2, 2016 - Hit the Grand for $14,214.91 playing 88 cents at CasaBlanca Casino in Mesquite, Nevada. (Hit it on a different machine than the one I won on in March.)

I wish it did show what was under the coins that you don't select, because I have hit the Mini and the Minor far more often than the Major or the Grand and always wonder.

But I have the tax forms to prove that you can win the Grand with far less than the maximum bet of $8.80 a spin.

Answer: Thanks for the kind words. And congratulations on your good fortune with 88 Fortunes.

The most recent question about 88 Fortunes was about how the software could force a particular outcome during the bonus round. The lady who asked the question assumed, as I once did, that it was possible to win each progressive because there are 12 coins displayed, four progressives and three matched coins needed to win a particular progressive. Four times three equals 12. Makes sense, right? But the machine does not reveal the coins you didn't pick, so we don't really know what is under those coins.

She guessed that the software constantly rearranged the progressives under the coins in order to force a match on the pre-determined outcome. None of that is necessary because the unpicked coins are not revealed. The software could have just chosen a sequence to reveal, like Mini Major Grand Grand Mini Major Mini.

After I had played the machine myself and received more emails from other players, it became clear that each progressive was NOT equally likely to be selected in the bonus. Everyone said that they almost always get one of the two lower progressives and only rarely get one of the two larger ones. I expected a more even distribution if each progressive had been equally likely.

So one lesson learned -- or at least inferred -- is that it's not possible to win each progressive each trip to the bonus.

Another fact reported by players is that they, like yourself, won the Grand without playing a max bet.

One of my contacts said that it is possible to win any progressive with any size bet, but would not elaborate on the algorithm used by the software. My guess is that the software randomly selects which progressive to award when the bonus round is triggered. The higher the progressive, the less likely it is to be chosen. It's also possible that different probability sets are used for each bet size, so the higher progressives might be more likely with higher bets.

After the progressive is chosen, the software populates the screen with two coins for each of the progressives not chosen and six coins for the progressive that was chosen. It then waits for you to reveal three matching coins.

It's also possible that the software could randomly choose one of three layouts. One layout is like the one above with the Mini forced. A second layout could have four coins each for Mini and Minor, and two each for the others. The third layout could have three coins for each progressive. This way, the software could either force the Mini or let your luck determine your prize from two or four possibilities.

Slot designers like to keep the math simple, though, so I think the most likely implementation is that the software chooses the progressive and then forces it by making it the only progressive that has three matching coins.

Question: I have heard many times that a Royal Flush happens close to every 40,000 hands.

I submit that is a wrong mathematical interpretation. One in 40,000 is the odds of RF. The frequency of RF does not need to follow the odds. Thus, anyone may not see a RF for a long time or a RF may hit several times in a session play.

The number of odds and frequency may come close as the number of play increases.

Answer: Well, no, odds and frequency are two different ways to state the probability of an event occuring. But I will grant you that saying that a royal flush occurs every 40,000 hands is leaving out an important phrase.

Royal flushes do occur about once every 40,000 hands on the average or in the long run.

Any set of 40,000 hands may contain 0, 1, 2 or any number of royal flushes. But if we add up the sets, the ratio of the number of royals over the total number of hands will approach 1 as we add in more sets -- which is what you said in your last sentence.

Still, odds and frequency describe the same value.