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We have $n$ people: $\alpha n$ are boys and $(1-\alpha)n$ are girls. They are standing in a line in a random order. We pick up one boy also at random.

What can one say about the probability that there are more girls than boys before this randomly selected boy if $n\to \infty$?

Is it true that this probability is $O(1/n)$?

Edit:

Yes, i meant $\alpha>0.5$. What is the correct approach to find the coefficient before $1/n$?

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    No, because if $\alpha \lt \frac{1}{2}$ the probability goes to $1$.2011-04-26
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    You have a small issue that $\alpha n$ must be an integer, so $\alpha$ must be rational with $n$ a multiple of its lowest terms denominator. Recasting the question it becomes $b m$ boys and $g m$ girls, with $b$, $g$ and $m$ positive integers (and $b>g$), what happens as $m \to \infty$?2011-04-27
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    Henry, i agree but i will leave current setup2011-04-27

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