Question of the Day - 17 January 2023

[Editor's Note: Over to you, Bob Dancer.]

Neither.

A cycle of 40,391(the actual cycle for 9/6 JoB) does not mean that if you’ve played 39,391 hands without a royal,  you’re more than average likely to hit a royal in the next 1,000 hands.

A cycle that size means that on your very next hand, the odds are 40,390-to-1 against getting a royal flush. This is true whether you hit a royal on your last hand or you’ve gone 32 years since your last royal.

The concept of being so many hands “into a cycle” is nonsense. No mastermind or regulator counts your misses and does something to give you one when you’re due. 

If you’re playing Ten Play, yes, you have ten separate chances to get the royal on your next hand, but it doesn’t mean that your odds are 10 out of 40,391. Usually, you’ll hit zero royals, of course, but you could hit any number between zero and 10. The odds of getting 10 royals are roughly 1 in 650,000 against, but you’re more likely to get 10 royals than four, five, six, seven, eight, or nine royals put together!

That last sentence might seem a bit of a puzzle, so let’s look at it a bit more closely.

To get multiple royals in Ten Play, but not all 10, the easiest way is to start from four to the royal. That is, a suited AKQJ with any card other than the 10 of the same suit. You get such a hand every 2,825 hands (including the suited KQJT9 straight flush, which you should keep unless you’re playing Deuces Wild).

From this 1-in-2,825 hands, it’s basically 1 in a million to hit five royals, 1 in 56 million to hit six royals, 1 in 450 million to hit seven royals, etc. To make the math easy, if you include the difficulty of starting with four-to-the-royal, it’s basically 1 in 3 billion to get five royals, one in 150 billion to get six royals, one in 1.2 trillion to hit seven royals, etc. These are very small numbers in comparison with the 1 in 650,000 chance of being dealt a royal, which gives you ten of them all at once.

These numbers are calculated using the binomial theorem. It’s not too difficult using Excel, but explaining exactly how to use Excel to do it is a lesson for another day.