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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