But the bad beat was not the jackpot I had in mind when I came in to play that day. When I got there, first thing I looked at was the Monte Carlo board, to see if anyone had hit the Hearts Royal Flush, or if that progressive was still up over $10,000.
DL has separate Monte Carlo progressives for each suit on the Royal Flush and straight flush "Steel Wheel" jackpots. They increase a couple hundred bucks a day, until somebody hits it, then they reset to $2,000. Most times they get hit at around $5K, or less, but occasionally one of them gets stubborn, holds out, and climbs up high enough to get everyone's attention.
And that was the case on January 1, when the Hearts Royal was up over ten thousand. Thing is, STILL AIN'T NOBODY HIT THAT SUCKER, and right now the Royal Flush Monte Carlo board is:
- Spades: $ 2,240
- Hearts: $ 28,526
- Diamonds: $ 2,888
- Clubs: $ 4,136
So when Tommy Fastplayer open limps on the button for the first time in 15 years, doesn't want to just win the blinds, wants to see the flop, what he got? Broadway Hearts? Ya think?
Couple years ago my Queen Ten of Hearts flopped a Royal, and I posted about how relatively easy it seemed to figure the probability to FLOP it like that: one time out of-
50/3 x 49/2 x 48/1 = 19,600
Ok, fine. But what if you don't flop it? What if you turn or river it, or just flop one, and need runner-runner? What's the overall probability for suited broadway holecards to just MAKE a royal, any which way you can?
Where I didn't break a sweat figuring probability to flop it, that overall probability thing made my head hurt. You've heard that a man should know his limitations? I found mine right there, and spent a couple years just thinking about it now and then, trying to find a way to approach it.
Hearts Royal being way up there right now got me thinking about it again, and Saturday night it hit me. As is often the case, the solution turned out to be not so complicated after all.
Turns out that flopping it is really no different than all those other ways to make it! Why did it take me forever to realize that? Why did I make it more complicated than it needed to be?
You need for 3 particular cards to land in any of 5 positions on the board. If they land in spots 1, 2 and 3, well, awesome, you flopped it. Good job.
But having those 3 cards land in any other combination of the five spots should have that same probability. That was my epiphany last Saturday. So I made this list of EIGHT ways your Royal cards can appear the board:
1. x x x o o flop it
2. x x o x o turn it
3. x x o o x river it
4. x o x x o turn it
5. x o o x x runner-runner
6. o x x x o turn it
7. o x o x x runner-runner
8. o o x x x runner-runner
9. x o x o x MISSING
10. o x x o x MISSINGThen I went from there, thinking that suited broadway should MAKE a royal EIGHT times out of 19,600.
A Humbling Experience
David H. happened by, and I showed him my work. He took a quick look, thought about it for three seconds, and said, politely, "There should be 10 ways to make it, not just 8."
He saw 3 royal cards landing in any of 5 board positions as a combinatorial, 3C5=10, which it is. I know how to spell that, but don't know the math well enough to do it, so I just too quickly made my sloppy, incomplete list of [what I thought were the 8] ways to make the royal. Should have been more careful.
Then David looked at my list of 8 and IMMEDIATELY told me 9 and 10, the two I had missed. Recognizing and working with patterns like that is a whole 'nother kind of measurable intelligence, and all this goes to explain why he's a professional, and I'm an amateur.
OK, so now we know that suited broadway holecards will make a royal TEN times in 19,600, or once in 1,960. Next question: How often are we dealt broadway hearts?
Easy. (52 x 51)/2 = 1,326 possible starting hands, and ten of those are broadway hearts. So you'll get them, on average, once every 132.6 deals.
So then the chance for you (or anyone) to be dealt broadway hearts AND make the royal, on the next (or any) deal, should be:
1/132.6 x 1/1960 = 1/259,896
Sure, I could have looked this up on Wizard of Odds, or some other website. But that's no fun.
NOTE: You can make a royal with one (or zero) cards in your hand and four (or five) cards on the board. That doesn't pay any jackpot, and was not considered at all in this discussion.
