Racing certainties: solution
If we discount the favourite (with odds of 1 to 1) and place none of our stake money on him then we are really betting on a three-horse race where Q is equal to
![\[ Q=\frac{1}{4}+\frac{1}{8}+\frac{2}{5}=\frac{31}{40}<1, \]](/MI/plus/issue34/outerspace/solutionhtml1/images/img-0001.png)
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of our stake money on runner 1, 
on runner 2, and 
on runner 3 we are guaranteed a minimum return of 
of our total stake in addition to our original stake money!
In the case of 
and an infinite number of runners (
) we have
![\[ Q=\sum _{i=1}^{N}\frac{1}{a_{i}+1}=\sum _{i=1}^{\infty }\frac{1}{i(i+2)}= \frac{3}{4}. \]](/MI/plus/issue34/outerspace/solutionhtml2/images/img-0003.png)
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