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Topic: counter-intuitive fact from everyday mathematics
Replies: 17   Last Post: Jun 24, 2012 10:11 AM

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Posts: 92
Registered: 6/21/12
Re: counter-intuitive fact from everyday mathematics
Posted: Jun 21, 2012 7:15 AM
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On Thursday, June 21, 2012 4:59:47 AM UTC-4, Paul wrote:
> 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

I fail to see how the result is counterintuitive. If anything, what is counterintuitive is that the probabilty is as high as it is. If a person has any experience at all with games involving dice then they would know that a double 6 is not all that rare. A typical game of Monopoly would feature several double sixes over the course of the game. 500 rolls without a double six would clearly be a run of luck of epic proportions. I can't speak for the mythical "person on the street" but I would imagine that most people wouldn't be surprised that rolling 500 pairs of dice and never getting a double six is somewhat akin to winning a lottery.

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