And, while we are on coincidencdes, I wrote (see below at end):

> 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").

Well, I don't give up that easily ... a bit of experimentation
gave rise to:

  (6*5*4*3*2*1)/1100377 = 0.000654321

and, to round it off nicely, 1100377 is a prime number. So there!
Ted.

On 14-Apr-11 21:17:30, Moore, Robert wrote:
> 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.
> 
> --------------------------------------------------------------------
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