Probability and Statistics

Forum Archive : Probability and Statistics

How many games to decide who's better?

From:   Stephen Turner
Date:   21 March 1997
Subject:   Re: Statistically speaking..

Dodd wrote:
> I am sure that this question has been asked many times before, but since
> I just found this newsgroup I would like to ask it again.
> Statistically, how many games of backgammon must two people play before
> one can definitively say they are the better player?

Unfortunately, it's not quite as simple as that. It depends how far you are
apart after those games. The standard way to set it up as a statistical
problem is to play a pre-determined number of 1-point games -- let's say
100. If you are the same standard, there's only a 2.5% chance that you will
win at least 60 of them, which is enough to reckon that you are probably
the better player.

Three further comments though:
1) If you are only a tiny bit better, this test is unlikely to pick up the
difference. In that case, we say that the test isn't very powerful.
Conversely, if you are much better, the test will usually pick up the
difference, and it's said to be a powerful test.
2) There's a thing called a "sequential test" whereby you don't have to
play the full 100 games if someone is way ahead after (say) 50 of them.
This is often used in medical experiments, for obvious ethical reasons. But
it's rather complicated to work out what the criterion for stopping and
deciding should be with a sequential test.
3) I assumed 1-point games. The real skill is in the cube, and I've also
neglected gammons. Unfortunately, it's not possible to work out an exact
statistical test for money games without knowing what proportion of each
type of score you would normally get between two equal players. For match
play, however, you're OK: if you play 100 5-point matches, instead of 100
1-point games, and just worry about the winner not the final score,
everything else works the same way.

That's probably more than you wanted to know. But I wouldn't want to
mislead you with an incomplete answer. :)

Stephen Turner
  Statistical Laboratory,  16 Mill Lane, Cambridge, CB2 1SB, England
       "This store will remain open during modernisation. We apologise
        for any inconvenience this may cause" Topshop, Cambridge
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