# odds

## Zero-scoring cribbage hand

Martin writes:

I am an avid cribbage player and have enjoyed the game for close to fifty years. I've never had a perfect twenty-nine hand but I've had three twenty-eight hands. I know that the odds of getting the twenty-nine hand are approximately one in a little over three million [ Actually, it's one in about 200,000 - Ed ]. Recently something very strange occurred to me. I dealt the first hand of a new game and here is how the scoring went:

The cut was not a jack

The laying down phase of the hand yielded the mandatory one point on a "go"

My regular hand yielded zero points

My crib hand yielded zero points

I have never heard of anyone else accomplish this feat. As you are aware the dealer of a hand will always score at least one point on a "go" or a "last card". So I am wondering - what are the odds of such an event occurring? I'm pretty decent with math problems however I'm not sure I even know exactly what equations are required to calculate the odds of getting a perfect "imperfect" hand. Can you help me figure out the end result?

To work out the odds of a zero-scoring cribbage hand (apart from the mandatory one point for go), we need to calculate how many such hands there are, and then divide that into the number of all possible cribbage hands. It is a tricky problem because we need to make no points in the play, assuming correct play. Anyone care to tackle it?

## 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

Cribbage rules and cribbage strategy make the discard one of the key elements of skill in cribbage. You must try to maximise the remaining points in your hand, while leaving yourself useful cards to play in different tactical situations during the pegging, and without giving your opponent cards which may help her in the crib. When discarding to your own crib, you will be trying to anticipate what kind of cards your opponent is likely to give you, and discard cards which will work with them to create big scores in the crib.