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.