I think Kevin wins on this.
Best wishes
Jane Galbraith

> I'm not sure if this is right. I mean, it clearly is if the result
> 1,2,3,4,5,6 means that the first dice (predefined in some way --- and I've
> given up using "die" as the singular of "dice") comes up 1, the second
> comes up 2 and so on. But I suspect that may not be what Robert meant. He
> might instead have meant that the scores were 1,2,3,4,5,6 in some order
> regardless of which dice was 1, which 2, which 3 and so on --- in other
> words that the set of scores was {1,2,3,4,5,6} regardless of order. In
> that case all possible sets of scores are not equally likely.
>
> The key point is that repetition is possible. You can get 1,1,1,1,1,1 as
> an extreme case. Also, I hope clearly, {1,1,1,1,1,1} is considerably less
> likely than {1,2,3,4,5,6}. The result {1,2,3,4,5,6} corresponds to 6! =
> 720 possible ordered results, but the result {1,1,1,1,1,1} corresponds to
> only one ordered result.
>
> In fact, unless I've got this wrong (which I probably have, was never very
> good at combinatorics), the unordered result{1,2,3,4,5,6} is more probable
> than any unordered result that includes repetition of one or more numbers.
>
> It might be easier to see with tossing two fair coins. The ordered
> possibilities are HH, HT, TH, TT. These are all equally likely. But if
> what you're itnerested in is not the ordered results, but just the
> probabilities of the 3 unordered possibilities {H,H}, {H,T}, {T,T}, then
> the middle one, i.e. the result that one (unspecificed which one) of the
> coins is a head and the other is a tail, has a probability of a half, and
> the other two are less likely, each having probability 1/4.
>
> Arguing from the dice (or coins) to the lottery doesn't really work, not
> in detail anyway, because in the (UK, and similar) lottery, you can't have
> the same number coming up twice in one draw, becasue they sample without
> replacement, but with dice, you can have the same number coming up more
> than once in a set of 6 throws
>
>
> Kevin McConway
> Professor of Applied Statistics
> Department of Mathematics and Statistics
> Faculty of Mathematics, Computing and Technology
> The Open University
> Walton Hall
> Milton Keynes MK7 6AA, UK
> Phone: +44-1908-653676
> Fax:      +44-1908-655515
> email:   [log in to unmask]
> ________________________________________
> From: John Whittington [[log in to unmask]]
> Sent: 14 April 2011 09:13
> To: [log in to unmask]
> Subject: Re: statistical trivia
>
> 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?
>
> As I'm sure you know, no, it is no more or less likely than any other
> outcome.
>
> This is, in fact, the very technique I use to try to make people (who
> think
> they may win!) realise how unlikely it is that they would win the
> Lottery.  First of all, I get them to understand and agree the point that
> 'random' means that ANY Lottery 'set of numbers' is equally likely to come
> out of the machine.  Most seem to find it quite easy to agree with that.
>
> I then ask them how likely they think it is that the Lottery draw would
> result in (1,2,3,4,5,6) and they generally say something along the lines
> of
> "virtually impossible".  It then ought not to be too difficult (although
> some can resist!) to get them to realise that it is therefore equally
> 'virtually impossible' _their_ choice of 6 numbers  will win!
>
> However, I bet you couldn't do it again :-)
>
> Kind Regards,
>
>
> John
>
> ----------------------------------------------------------------
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