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Playing in the District 6 NAP qualifier over the weekend, you are sitting East on what seems at first glance an unremarkable hand: 

West
J75
874
87
KQ843
North
AQ
J632
AJ4
A762
East
10943
AK105
K1096
10
South
K862
Q9
Q532
J95
W
N
E
S
1NT
P
P
P
D
1NT North
NS: 0 EW: 0

You plunk down the A and partner encourages. You cash the K and play back your low heart, partner's 8 forcing out declarer's J. Declarer unblocks the AQ, and leads a low diamond towards dummy. You duck and dummy's Q wins. Declarer cashes the K and the A, then shrugs and exits with the J. You cash your winners and declarer gets the last trick with the A. Making 1 for a 60% board for E-W when a few declarers went down in game and a few more managed an 8th trick in 1NT. Really doesn't seem like there's much to this hand. So why is this worthy of an article?

West
J75
874
87
KQ843
North
AQ
J632
AJ4
A762
East
10943
AK105
K1096
10
South
K862
Q9
Q532
J95
W
N
E
S
1NT
P
P
P
D
1NT North
NS: 0 EW: 0

Well, at the point you win the K you are on lead holding:

10 10 10 10

Not only do you have a 4-of-a-kind to end the hand, but three of them are winners! Pretty sure I've never seen this before, although it does seems like something you could easily miss if you weren't paying attention. And it almost didn't happen here - declarer had to recognize that he couldn't lose his A and might gain on a squeeze or endplay, particularly if the defense wasn't perfect.

So just how rare is this? The odds of holding any 4-of-a-kind to start the hand are around 30 to 1 against, so it seems ending with a 4-of-a-kind must be extremely unusual. If we assume that all play orders are equally likely for a moment, the chances of ending with your 4-of-a-kind are (by symmetry) the same as the odds of your first four cards being your 4-of-a-kind, namely (4/13 * 3/12 * 2/11 * 1/10), which comes out to 1 in 715. That would make the overall odds around 1 in 20000. To put that in perspective, it's a bit more rare than being dealt a 7-6 hand or an 8-high hand (per Pavlicek's Odds and Theory page). If we specify that you must end up on lead with 4-of-a-kind that would make it at least 4 times less likely.

But that's assuming all play orders are equally likely, and I suspect this is even more rare if we take play order into account. Yes, you might have the choice at some point to preserve your 4-of-a-kind for funsies, but you probably won't notice and exercise this choice until you're down to 5 or 6 cards at most (and maybe not even then!). Set against that:

  • Many hands are claimed before you get down to 4 cards
  • Intuitively, it feels like coming down to one card in each suit would be less likely than one would expect a priori, particularly in a suit contract. That being said I can't think of a good way to test this and this is the type of intuition that humans think we're much better at than we actually are.

Anyone else ever seen this before? 

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