My two most recent posts here have been about a story reporting a coincidence as more exceptional that it is and ‘bad maths’ reported in the media. Both are examples of mathematical stories being reported in a way that is not desirable. Somehow, though, I like the whist story and dislike the PR equations. I have been thinking about why this might be the case.
The PR-driven, media-friendly but meaningless equations from the first article are annoying because they present an incorrect view of mathematics and how mathematics can be applied to the real world. Applications of mathematics are everywhere and compelling, yet the equations in these sorts of equations seem to present little more than vague algebra. The commissioned research with seemingly trivial aims I find more difficult because, as commenters on that article pointed out, it is really difficult to decide what is trivial. Still, reporting that a biscuit company has commissioned research into biscuit dunking is either meaningless PR or else a matter of internal interest, and certainly nothing like what I expect mathematicians do for a living.
Coming back to our Warwickshire whist drive: what do I like about this story? It too presents incorrect information about mathematics and the real world, claiming that the event, four perfect hands of cards dealt, is so unlikely that it is only likely to happen once in human history (and it happened in this village hall!).
I think the difference is that the mathematics used, combinatorics and probability, appear to be correctly applied. The odds quoted, 2,235,197,406,895,366,368,301,559,999 to 1, are widely reported and I see no reason to doubt them.
The problem, then, is one of modelling assumptions. Applying a piece of mathematics to the real world involves describing the scenario, or a simplified version of it, in mathematics, solving that mathematical model and translating the solution back to the real world scenario. In this case, the description of the scenario in mathematics assumes that the cards are randomly distributed in the pack. This modelling assumption, rather than the mathematics, is where the error lies.
The result is still a bad maths news story, presenting a mathematical story as something other than what it is, but while the PR formulae are of little consequence, this incorrect application of a correct combinatorial analysis is something we can learn from.
I wonder how perfect the shuffling was.
If you consider a valid hand to be more ‘structured’ than an invalid hand then wouldn’t one expect the deck to become more structured over time if the shuffling is anything less than perfect?
Playing the game would lead the deck to become structured as players construct hands. Imperfect shuffling would not remove all of this structure.
As the deck becomes more structured then my intuition tells me that it would be more likely to be dealt valid hands than if the deck were perfectly random.
I wonder how much this affects the probabilities. I wonder if it can be quantified. I wonder how hard it would be to simulate.
Cheers,
Mike
I’d suggest that bad maths stories are quite useful, as long as there’s some sort of reply mechanism to let people point out problems.
Whether 1=0.9.. , the Tuesday Child problem and Monty Hall are staples of chat fora (to the point of being banned topics on some), and it’s good to see both correct proofs and faulty, subtle ones. People put them forward, they get picked to pieces, and the best are adopted. It’s a very similar process to academic publication and review, just on a more amateur (and more accessible) level.