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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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scattered

Posts: 89
Registered: 6/21/12
Re: counter-intuitive fact from everyday mathematics
Posted: Jun 21, 2012 3:25 PM
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On Thursday, June 21, 2012 1:51:15 PM UTC-4, Phil H wrote:
> On Jun 21, 7:10 am, pepste...@gmail.com wrote:
> > On Thursday, June 21, 2012 12:15:43 PM UTC+1, scattered wrote:
> > > 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.
> >
> > Here's one that's interesting (to me).  And I don't actually know the answer.
> > People keep doing the 500-dice-roll game until eventually someone wins the game and gets a 66-free 500 roll set.
> > What is the probability that this winner got a total number of 6's among all 1000 dice that was less than expected (in other words 166 or less)?
> >
> > It's an interesting one because, without bothering to think about it much, the answer is obviously > 50% but I really haven't got much of a clue beyond that.  Nothing inside the 60% to 99.999999% range would particularly surprise me.
> >
> > Paul Epstein- Hide quoted text -
> >
> > - Show quoted text -

>
> Its a conditional probability problem with a binomial
> distribution.....given that a 66 doesn't occur, what is the
> probability of getting at most 166 6s when rolling 2 dice 500 times?
> The probability of getting one six on a single roll of 2 dice is now
> only 10/35 = 2/7. So, the probability of getting at most 166 is
> binomcdf(500,2/7,166) = .9896. This is a cumulative calculation on a
> TI-83 of all 167 outcomes given that a single outcome (say 150 6s)is
> calculated as 500 nCr 150*2/7^150*5/7^350.
> Phil H


Nice, simple solution.



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