Paul
Posts:
258
Registered:
7/12/10
|
|
counter-intuitive fact from everyday mathematics
Posted:
Jun 21, 2012 4:59 AM
|
|
There are a few iconic "surprising facts" from elementary mathematics
which are often talked about at high-school level. A few that spring
to mind are the birthday stuff, the let-the-penny-double (or any power-
of-2 equivalent), and the fact that if you shuffle a set there's a
probability of 1-1/e that at least one element in the shuffled set is
in the correct position. (This is referenced as shuffling cards, or
an absent-minded postman etc.)
I'd like to add my own example to this famous list. I wouldn't be
terribly surprised if someone has already discussed it, but I've
certainly never heard it referenced.
If you roll a pair of unbiased dice 500 times, the probability that
you never get a double six is less than 1 in 1.3 million.
I think that 1) To the "person in the street", this is even more
surprising than anything in the standard list. I think most people
would be delighted to spend a dollar on rolling a pair of dice 500
times to be rewarded with a million dollars if they miss 66 each
time. But this would not be good odds.
2) This fact has the advantage of being true both theoretically and
practically, with almost no necessity for discussion about the extent
to which real-world practice matches theory.
In that sense, it is a "cleaner" example than anything in the standard
list above. Everything in the standard list has possible real-world
objections. With the birthday stuff, it's a debatable assumption that
births are uniformly distributed throughout the year. With the
shuffling stuff, approximation is used and other assumptions made (the
answer is actually rational, not 1 - 1/e). The power-of-2 stuff often
raises issues of meaning, depending on how large the power of 2. (You
can't literally fold a piece of paper 1000 times. It's unclear what
it means to talk about a sum of money of £10^50 etc.)
Did I come up with a decent example? Or is it something that's been
given many times before without me being aware of it?
Thank You,
Paul Epstein
|
|
