## 27 November 2011

### Four "perfect hands"

From a story in The Sun, via Arbroath:
A group of whist players were each dealt a complete suit in an opening hand — beating odds of two thousand quadrillion to one. Wenda Douthwaite, 77, and her three friends were "gobsmacked" when they were dealt the hand during a game last week. Mathematician Dr Alexander Mijatovic, a probability expert at Warwick University, worked out the odds as being 2,235,197,406,895,366,368,301,559,999 to 1...

Everything was done as usual. The pack of cards was an old one. The cards were shuffled, cut and dealt as normal... "We play regularly and are always very careful to make sure the deck of cards is shuffled repeatedly.
I don't mean to sound churlish here, because they surely deserve their moment of fun and fame, but mathematically every other set of hands they were dealt that night would have had the same probability of occurring.  This arrangement of cards is visually striking, but not mathematically any less likely than any other specific distribution.

1. Allow me to top you in churlishness (to your credit, not a high hurdle) - if you were to play a billion hands of whist and bridge a day, it would, on average, take 164,000 times the age of the universe to do what they said they did. Either they're lying, someone played a complicated trick on them (I'm going with the guy on the right) or we should start looking for other signs of the coming Armageddon.

2. Yeah, the guy on the right dealt.

3. I found a different number elsewhere -

Number of possible deals = 52!/(13!)^4 = 53,644,737,765,488,792,839,237,440,000

(source: http://www.bridgehands.com/P/Probabilities_Miscellaneous.htm)

But I would still argue that this set of four hands is no less probable than any other specific set of four hands. Sit down, deal four hands. The odds of those hands being dealt would be 1 in 53,644,737....... etc. Just because something is mathematically "improbable" does not make it unlikely to occur. (I think I learned that from the Hitchhiker's Guide to the Universe).

There are several math enthusiasts among the readers here; we'll see if they pick up on this question.

4. Bub, great minds again...I was thinking the same thing. (When I was a kid I had a craze for poker for a couple of months. I'd play with my father, and once when he got up to make a sandwich, I stacked the deck, and when he came back, I dealt him a royal flush. He was astounded--genuinely--and only got suspicious when I couldn't keep a straight face.)

I also had the same thought about Armageddon. That wonderful sentence from Arthur Clarke's story "The Nine Billion Names of God" flashed through my mind: "Overhead, without any fuss, the stars were going out."

5. @Minnesotastan I agree with your analysis. Any set of four bridge hands published in the bridge column of your local paper are just as unlikely.

This very interesting radiolab segment discusses the need for large combinatorics in games to keep them interesting.

6. I figured out the source of the discrepancy.

The odds of choosing four hands of 13 cards from a 52 card deck are in fact:

one in
(52 choose 13) * (39 choose 13) * (26 choose 13) * (13 choose 13) = 5.36 x 10^28

This agrees with the website you found. However this is the probability of dealing (for instance) North all the spades, East a all the diamonds, South all the hearts, and West all the clubs.

If you don't care which seat gets which suit you may divide by 4! = 24 which yields the answer given by Dr. Alexander Mijatovic in the quoted article.

7. 2.35 x 10^27

8. Thank you, Dan.

9. But all those other hands, although equally likely, are nowhere near as special in terms of the mechanics of the game, not to mention the aesthetic value of this arrangement. Though I know the mathematicians are right, I still can't help but see this as a freak occurrence. Possibly because the formal aspects of this hand align so perfectly with those of the game, i.e. four players, four suits, and just enough cards per hand to accommodate a whole suit? Maybe that's why this hand 'seems' so perfect.

10. The players may be very confident that the cards were shuffled perfectly, but they must admit to a small chance of error. Whatever the chance of an improper shuffle, it is surely much greater than the chance of dealing four perfect hands. The only reasonable conclusion, assuming that the players are telling the truth, is that the cards were not shuffled properly.