Probability of great hands

10 August 2017

By John Grochowski

QUESTION: I got three-of-a-kind two hands in a row in Three Card Poker. It was pretty exciting. I was betting $5 on Pair Plus, so at 30-1 I won $150 on the first time. Then I raised the bet to $10 and won $300 the second time. That’s all in addition to wining ante-play against the dealer, too, and getting the 4-1 ante bonuses there.

A couple of questions about it. What are the odds of two threes-of-a-kind in a row? How would that compare to two straight flushes in a row? While we’re at it, how about two in a row in some other games – two blackjacks in a row, two of the same number in a row in roulette, two royals in a row in Caribbean Stud?

ANSWER: Congratulations on a great win!

You’ll see three of a kind at Three Card Poker an average of once per 425 hands. The find the probability of two in a row, you need to square that: 425 times 425 is 180,625. So your two in a row was a 1 in 180,625 shot.

That’s a long shot, but long shots happen in casinos every day. There are so many wagers made that things that seem improbable are not only possible, but inevitable.

For comparisons to other wagers, we follow the same formula – find the chances of the event happening once, then multiply that chance by itself.

In Three Card Poker, straight flushes make up 1 / 460th of all possible hands, so you chance of a straight flush on any given hand is 1 in 460. For two in a row, it’s 1 in 211,460.

Blackjacks are much more frequent at about 1 per 21 hands. The chance of two in a row is about 1 in 441 – narrow enough that you see consecutive blackjacks often. On double-zero roulette, each number is 1 of 38. The chance of the same number turning up twice in a row is 1 in 1,444.

With Caribbean Stud, we enter the world of extreme rarity. Caribbean Stud is a five-card stud game, and the chance of a royal per hand is 1 in 649,740. That puts two in a row at 1 in 422,162,067,600 – that’s 422 billion and then some.


QUESTION: Playing online, I found a weird game called Mini-Roulette. It has one zero along with numbers 1 through 12.

What’s the house edge? Do you think we’ll ever see it in live casinos?

ANSWER: I’d never heard of the game, so I searched and found Michael Shackleford’s analysis at wizardofodds.com.

Winning bets pay 11-1 on single numbers, 5-1 on two-number splits, 3-1 on three-number bets that include rows as well as 0-1-2 and 0-2-3, 2-1 on four-number columns or corners, and even money on red, black, odd, even, 1 through 6, 4 through 9 or 7 through 12.

In addition, you get half your losing bet back if the ball lands on 0.

That last bit is important. It means the house edge is less on bets that don’t include 0. House edges are 3.85% on all bets that don’t include 0 and 7.69% on those that do.

I would not expect to see live-action versions of this game come to brick-and-mortar casinos.

Properly balanced wheels are expensive to build and maintain, and it seems like Mini-Roulette would just draw customers from existing roulette tables rather than create new business. The game’s best shot at a breakthrough in live casinos would seem to be inclusion on multi-game electronic tables.