Odds of a 29 hand in cribbage

It is possible to work out the exact chances of getting a 29 hand in cribbage. Mathematician and stats expert David desJardins explains:

You need to be dealt three fives, the jack of the fourth suit, and two other cards neither of which is a five. The total number of such six-card hands is 4*(47*46/2) = 4324, out of (52*51*50*49*48*47/720) = 20358520 possible hands. Given this event, the probability of turning up the fourth five is 1/46. So the probability is:

4324 / 20358520 / 46 = 1 / 216580 (very roughly, 200,000 to 1)

Cribbage master Michael Schell elaborates on this argument, and the corresponding odds in the 3- and 4-handed games, in this Cribbage Forum article. He also notes:

The 1 in 216,580 figure jibes well with the actual incidence rate of 29 hands in sanctioned tournaments in North America. The ACC pays $100 for a 29 hand received in sanctioned play, and thus publishes a "Club 29" list each season. To be exact, the incidence is a tad lower than the odds predict, since the odds assume you keep an eligible hand (5-5-5-J) whenever you can. Since you wouldn't always want to do this (defending in an endgame for example), the actual occurrence of 29 hands among experts will be a bit less frequent than the mathematical calculation predicts.

Are the odds of a 29 hand the same in a 3-player cribbage game?

They are much longer. Michael Schell again has the complete proof, but the short answer is 1 in 649,740.

This is a result of only being dealt five cards, so there are much fewer "potential 29s" to choose from. The same odds apply to a four-handed game, or two-player five-card cribbage.

How many possible ways are there of making a 29 hand in cribbage?

The 29th point comes from the nob Jack, and since you have all four fives, any of the four Jacks will do. Thus there are four possible 29 hands in cribbage, with the Jack of each of the four suits being the turn up card.

Go to the main 29 hand page

Even less likely than that.

These odds don't appear to take into account how people cut for the Starter Card. Most people cut near the middle of the deck. It's pretty unusual for someone to cut just the top 2 cards, for instance.

So, the 5 Starter Card generally needs to be in the "sweet spot" near the middle of the deck. If it's the top card or the bottom card, it cannot be cut at all. This fact makes it even less likely than the odds stated above, but I don't know how such a "flexible" event can be calculated.

Oops!

I retract that last comment. The odds are unaffected by where the deck is cut. My bad.

Cannot cut just anywhere

According to the rules you must leave at least 4 cards on the top of the deck or 4 cards on the bottom when you cut. You cannot cut the cares leaving 2 cards on the top of the deck. That is not a proper cut. www.cribbage.org

a 29 point hand is not having

a 29 point hand is not having all four 5s and a Jack turning up. It is having three 5s and a Jack with the 5 that matches the Jack's suite turning on the cut. I think this actually decreases the odds from the stated numbers above.

odds for a 29 hand

the odds seem to be based on just one hand being dealt. there should be 12 cards missing from the deck when you cut, and you can't use the top or bottom 4 cards. Does this affect the odds.

And that's exactly what it

And that's exactly what it says. And the calculation is precisely correct. It's an easy-to-moderate homework problem for an undergraduate discrete math course.

The calculation is correct.

The calculation is correct. It's the comment at the bottom under the title "How many possible ways are there of making a 29 hand in cribbage?" that doesn't make sense.