# What are the odds?

Several years ago, I played at a \$3-\$6 limit table (Texas No-Fold 'em) and on one hand, five players stayed in after the blind was raised and capped. The flop came A-3-7. More bets, no folds. The turn was a 9. More bets, no folds. The river was a Q. More bets, no folds.

When the cards were turned over, the hands were pocket aces, pocket 3s, pocket 7s, pocket 9s, and pocket queens.

Five players, five pairs, and all hit their sets. What are the odds?

Adding to the intrigue, there were no straight or flush possibilities, thus my pocket aces took down a nice pot (for a low-level limit game).

Thanks!

• A pity you did not have a screenshot of that. I'd like to see that picture xD Commented Jul 20, 2016 at 3:46
• I would say the odds in you making this story up are considerably higher than the odds of this occurring haha. Commented Sep 8, 2016 at 15:21
• Foxes: No, true story. The dealer was Danny at Spirit Mountain Casino in Oregon. Commented Sep 12, 2016 at 20:48
• You could improve the title to something like "What are the odds of 5 players and 5 sets". Not a big issue though.
– user4555
Commented Sep 14, 2016 at 18:03

From a probability perspective, you can think of this as two events: the first being each of the five player getting dealt a pocket pair, and the second dealing five cards that match those pocket pairs exactly.

First let's begin with the probability that there are 5 different-suited pocket pairs dealt in one hand (note that 78 / 72 / 66 / 60 / 54 are the number of available pairs (with distinct rank) left in the deck before/after each hand is dealt, starting with the first hand and ending with the fifth):

( 78/(52 choose 2) ) * ( 72/(50 choose 2) ) * ( 66/(48 choose 2) ) * ( 60/(46 choose 2) ) * ( 54/(44 choose 2) ) = .000000669415

Next, we find the probability that EACH OF THE FIVE pocket pairs is matched on the board, making five three-of-a-kinds:

.000000669415 * (2/42) * (2/41) * (2/40) * (2/39) * (2/38) * 5! = .0000000000251817

So the odds of this hand happening are just about 1 in 40,000,000,000.

Some other interesting events with better odds:

• Odds of dating a supermodel: 88,000 to 1
• Odds of becoming president: 10,000,000 to 1
• Odds of having identical quadruplets: 15,000,000 to 1
• Odds of winning the California lottery: 13,000,000 to 1
• Odds of contracting the human version of mad cow disease: 1 in 40,000,000
• Calling @paparazzi ;) Commented Aug 19, 2016 at 21:55
• As stated my answer is board alone hitting. Commented Aug 21, 2016 at 14:05
• Thank you, David C. That's pretty crazy. Since that hand, I've seen three sets go to the river a couple of times, but even that's rare. Commented Aug 22, 2016 at 22:14