Most typically, there are 2 teams with 2 players each (the composition of the team may change during the tournament which consists of something like 25 games per afternoon).
Sometimes men win but it seems that more often, women win, and so on. I don't want to go into that. ;-) Instead, I want to discuss an unlikely event and people's reactions to it.
A week ago or so, one of the players – whom I will refer to as HM – was the dealer who deals out 14 cards out of 108 Canasta cards to each of the 4 players. During the afternoon, there were two interesting games.
- In one of them, one player got 7 jokers-or-deuces
- In the other, another player got 8 jokers-or-deuces
Needless to say, I had to be nearly silent about this topic for hours. The remaining three players teamed up to laugh at me and claim that I was obviously a nutcase, that there could be no pattern and even if there were a pattern at some point, it will surely get lost, and so on, so the unlikely events above must be due to pure chance.
It's not hard to calculate the odds exactly – a particular combinatorial number – but it's faster to write a short Mathematica code and run a simulation. First, you should know the problem. There are 108 cards in the deck, 4 of them are "jokers" ("big jokers" in our terminology) and 8 of them are "2" or deuces ("small jokers" in our jargon). Big and small jokers are "wild cards" and comparably helpful to complete canastas. So in total, there are 12 wild cards in the deck of 108 cards. What is the probability that a random subset of 14 cards among the 108 cards contains at least 7 wild cards? And what about the odds for at least 8 wild cards?
Here's a quite Mathematica code one may write down in a minute:
Sorry, "zuliku" stands for "number of wild cards". Now, a player gets at least 7 wild cards in 1 in 14,000 attempts; he or she gets at least 8 wild cards in 1 in 300,000 games or so. Please free to calculate the exact numbers (or see the comments for the exact formula).attempts = 1000000;
Dynamic[{j, successes}]
successes = 0;
For[j = 1, j <= attempts, j++,
a = Table[RandomInteger[107], {i, 1, 14}];
While[Length[Union[a]] < 14,
a = Table[RandomInteger[107], {i, 1, 14}];];
zuliku = 0;
For[k = 1, k <= 14, k++,
zuliku = zuliku + If[a[[k]] <= 11, 1, 0];
];
successes = successes + If[zuliku >= 7, 1, 0];
];
Print[successes/attempts];
During the afternoon, there are 25 games and each of the 4 players may receive the "surprisingly high" number of jokes, so there are effectively 100 opportunities per afternoon. The probability that someone gets at least 7 jokers sometime in the afternoon is 1 in 140 or so. The probability that someone gets at least 8 jokers sometime in the afternoon is 1 in 3,000. If you demand both of these things to occur in different games in the same afternoon, you must pretty much multiply the odds. So I think that the chances of the 7+8 miracle is 1 in 400,000 per afternoon or so.
Even before I tell you that the 7 or 8 jokers for a single player occurred 3 times in recent weeks and not just twice, this is an extremely low probability, close to a 4.5-sigma certainty that it couldn't have occurred by chance. (It seems to me that I am very close to 5 sigma if I refine the hypothesis so that only the 25 games in which HM is the dealer count.) Even if you account for the look-elsewhere effect by acknowledging that the game has been played on hundreds of afternoons, the chances are still low and they suggest that the null hypothesis – pure chance – is contrived, to say the least.
Incidentally, I think that I also know the mechanism by which HM ends up with a higher success rate when it comes to her or his ability to deal out 7 or 8 jokers. She or he shuffles the deck of cards by combining two half-decks in such a way that (ideally) every even card is from the left half-deck and every odd card is from the right half-deck. They alternate. If your left half-deck has a large concentrations of jokers, there will be many jokers among the cards with even (or odd) IDs. If she or he shuffles the cards by this method once again, the concentration of jokers gets diluted by another factor of two and when the shuffling is done regularly enough, which is not insanely unlikely, the ordering of the (many) jokers will be equal modulo 4. Roughly speaking, every fourth card will be a joker. When HM deals out the cards to the players, one by one, and that's how she or he is doing it (just like myself), it's obvious that the same player may get all the jokers.
Feel free to discuss the technical issues, pros and cons, of these arguments and mechanisms.
But my main surprise is the broader sociological question: Why do the other players feel so certain that they think that they don't even have to do any calculation – they don't have to know any maths beyond the fourth grade of the elementary school, to be more precise – to generate a sensible judgment on such an obviously subtle, quantitative, statistical question? They must be clearly convinced that mathematics doesn't operate anywhere in the world. They must clearly be convinced that mathematics can't possibly help you to judge such questions. Cards must be perfectly random from God or another supernatural or divine agent even if you don't shuffle them. ;-)
Why? How does the belief work? How does the world in their picture work? I have lots and lots of empathy – perhaps even more empathy than Sheldon Cooper ;-) – but I just can't understand it. As a person who sees equations behind everything and who feels very humble if not detached whenever I can't solve or even estimate the relevant maths in a situation, I simply can't understand where the skyscraping certainty of people who have almost no clue and who haven't even tried to think about the issue quantitatively comes from.
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