An Observed Extreme Probability Event

[gmax137]

So, did that bridge hand with only even cards an artifact of the shuffling procedure or was it just luck?

Prestidigitators have long studied shuffle tricks, the famous reference being Chris Marlo's 1953 "Faro Notes." A perfect shuffle is dividing the deck into two equal halves then alternating cards from each half. Eight consecutive perfect shuffles will return a deck of cards to its original order. Decks of cards are always sold with the cards in the same order. Five perfect shuffles will partition a brand new deck into half even cards and half odd cards. My thesis is that in the 2005 Bridge World championship a brand new deck was shuffled seven times, an unusually zealous attempt at randomization. If the shuffles are perfect this will generate a deck where there is a sequence where every fourth card is an even card. When four hands are dealt this is will result in a hand with all even cards, such as was seen at the table.

The shuffles didn't have to be perfect, just close enough. There are also numerous sequences where errors cancel one another. Believe whatever you want, but I'll take this over the one in 2.4 billion shot that assumes a uniform distribution.

I don't know about Bridge, but in casinos I often see the dealers "washing" the cards, ie, mixing them up the way a kid who can't shuffle would.

 

[Stephen Tashi]

Like many probability questions on the forum that touch on real life situations, this discussion doesn't make much reference to mathematical statistics !

It's interesting to look at the question from the point of view of hypothesis testing - not that hypothesis testing clarifies the question, but rather that the question reveals a lot about the subjective nature of hypothesis tests.

It often happens in hypothesis testing that any particular observed outcome (e.g, any deal) has a very small probability - smaller than the traditional alpha values used in hypothesis tests. So the "acceptance region" for accepting the null hypothesis ( that the deal was randomly dealt) is defined in terms of some larger set of deals. (Some statisticians violently object to the terminology "accepting the null hypothesis". They prefer to look at the complement of the acceptance region, which is the rejection region.)

To specify the acceptance region (which the OP calls an equivalence class of hands) we must either enumerate all the deals in the set or state some some characteristic of a deal that determines whether it is in the set. The probability that a hand falls into an acceptance region can vary widely depending on how that acceptance region is defined - as illustrated by previous posts in this thread. Although statisticians want to be unbiased, it is hard to resist the temptation of trying out various acceptance regions containing the observed result to see whether the associated probabilities please one's intuition.

The textbook treatment of acceptance regions is restricted to "one-tailed" or "two-tailed tests". This is the case when the outcome is a numerical measurement. Intuition that those acceptance regions are good is convincing, but proof that they are optimal (in some sense) is hard to appreciate. It has to do with the "power" of a statistical test.

When it comes to defining the power of a statistical test on deals of cards, I don't see any objective way of doing it. In the case of a numerical measurement, one graphs a function of the difference between the two scalars. For deals of cards, does any (possibly multivariable) function suggest itself ?