My 9 year-old daughter’s soccer games are often high-scoring affairs. Double-digit goal totals are not uncommon.  So when her team went ahead 2-0 on Saturday someone on the sideline remarked that 2-0 is not the comfortable lead that you usually think it is in soccer.

But that got me thinking.  Its more subtle than that.  Suppose that the game is 2 minutes old and the score is 2-0.  If these were professional teams you would say that 2-0 is a good lead but there are still 88 minutes to play and there is a decent chance that a 2-0 lead can be overcome.

But if these are 9 year old girls and you know only that the score is 2-0 after 2 minutes your most compelling inference is that there must be a huge difference in the quality of these two teams and the team that is leading 2-0 is very likely to be ahead 20-0 by the time the game is over.

The point is that competition at higher levels is different in two ways. First there is less scoring overall which tends to make a 2-0 lead more secure.  But second there is also lower variance in team quality.  So a 2-0 lead tells you less about the matchup than it does at lower levels.

Ok so a 2-0 lead is a more secure lead for 9 year olds when 95% of the game remains to be played (they play for 40 minutes). But when 5% of the game remains to be played a 2-0 lead is almost insurmountable at the professional level but can easily be upset in a game among 10 year olds.

So where is the flipping point?  How much of the game must elapse so that a 2-0 lead leads to exactly the same conditional probability that the 9 year olds hold on to the lead and win as the professionals?

Next question.  Let F be the fraction of the game remaining where the 2-0 lead flipping point occurs.  Now suppose we have a 3-0 lead with F remaining.  Who has the advantage now?

And of course we want to define F(k) to be the flipping point of a k-nil lead and we want to take the infinity-nil limit to find the flipping point F(infinity).  Does it converge to zero or one, or does it stay in the interior?

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