
Re: counterintuitive fact from everyday mathematics
Posted:
Jun 21, 2012 3:25 PM


On Thursday, June 21, 2012 1:51:15 PM UTC4, 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 UTC4, Paul wrote: > > > > There are a few iconic "surprising facts" from elementary mathematics > > > > which are often talked about at highschool level. A few that spring > > > > to mind are the birthday stuff, the letthepennydouble (or any power > > > > of2 equivalent), and the fact that if you shuffle a set there's a > > > > probability of 11/e that at least one element in the shuffled set is > > > > in the correct position. (This is referenced as shuffling cards, or > > > > an absentminded 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 realworld 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 realworld > > > > 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 powerof2 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 500diceroll game until eventually someone wins the game and gets a 66free 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 > TI83 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.

