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Thanks to all - I didn't expect such interesting exchanges. Yes I just picked up all the dice at once and threw them into the bottom of a box. The numbers were indeed 'eye-catching' and my first reaction was Wow! that's remarkable. Then as I carried on tidying up I think I realised that almost any order would have the same chance (though I wasn't _quite_ sure). The eye-catchiness is probably the most interesting aspect of the outcome of the throw. Probably never do it again (unless, of course I get 6, 5, 4, 3, 2, 1 - joking this time). Robert Professor Robert Moore School of Sociology and Social Policy Eleanor Rathbone Building The University of Liverpool L69 7ZA Telephone and fax: 44 (0) 1352 714456 ________________________________________ From: email list for Radical Statistics [[log in to unmask]] On Behalf Of Ted Harding [[log in to unmask]] Sent: 14 April 2011 11:20 To: [log in to unmask] Subject: Re: statistical trivia Going back to Robert's original message: At 08:27 14/04/2011 +0100, Moore, Robert wrote: > Helping my grandchildren clear up last weekend I collected > six assorted dice and tossed them into the games box. > The dice landed 1, 2, 3, 4, 5, 6. > This looked like a quite remarkable outcome and my immediate > thought was that there must be huge odds against this. But > is the outcome of this throw any more or less likely than > any other? The statement of the the situation leaves something to the imagination, but on "plausibility" grounds I interpret it as describing that Robert picked up 6 dice and threw them all at once into the box. Then he noticed that they were all different. The chance of "all different" without distinguishing between the individual dice can be calculated as follows. For the sake of argument, suppose they are all different colours: Red, Orange, Yellow, Green, Blue, Indigo, and take them in "spectral order". The Red one will be whatever it turns out to be, P = 6/6. The Orange one has to be different from the Red, P = 5/6. The Yellow one has to be different from both Red & Orange, for which P = 4/6. And so on, so the the probability that they are all different is: (6*5*4*3*2*1)/(6*6*6*6*6*6) = 0.0154321 = 1/64.8 So, if I have correctly interpreted Robert's description, the "all different" outcome is not that unlikely. John wrote: "However, I bet you couldn't do it again :-)". In that case, then my apologies, Robert, for blowing your chances of a very profitable bet with John. However, if (as is possibly compatible with Robert's description) he picked up the dice one by one from various places and, as he picked up each one, he threw it into the box, and then observed that, in the order in which he threw them, he observed that they came up 1, 2, 3, 4, 5, 6, then that is very unlikely indeed: Prob = 1/(6^6) = 0.000021433 or 1/46656. This may be what John was thinking of -- in which case I think Robert could easily have got John to accept odds of 10000:1. Apologies again! This question raises a wider issue. Robert noticed the pattern "1, 2, 3, 4, 5, 6". There might be other patterns which could catch his eye -- "all equal" is one, of course, but that really is unlikely -- 1/(6^5) -- but say three equal to one value, the other three equal to another value, might also be eye-catching; I make the chance of this to be 100/(6^5) = 0.01286008 = 1 in 77.76, also not that unlikely. So the chance that Robert might observe some "coincidence" sufficiently eye-catching to prompt him to write to us could be quite high. Noticing a "coincidence" of some previously unspecified kind is, in real life, not at all unlikely! While I'm at it, I quite like the "54321" in the above result (6*5*4*3*2*1)/(6*6*6*6*6*6) = 0.0154321 (Pity about the preceding "1" though; I'd have preferred "6"). Best wishes to all, Ted. -------------------------------------------------------------------- E-Mail: (Ted Harding) <[log in to unmask]> Fax-to-email: +44 (0)870 094 0861 Date: 14-Apr-11 Time: 11:20:14 ------------------------------ XFMail ------------------------------ ****************************************************** Please note that if you press the 'Reply' button your message will go only to the sender of this message. If you want to reply to the whole list, use your mailer's 'Reply-to-All' button to send your message automatically to [log in to unmask] Disclaimer: The messages sent to this list are the views of the sender and cannot be assumed to be representative of the range of views held by subscribers to the Radical Statistics Group. 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