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100 small bags of coins are placed in a large cauldron. The 99 of these bags that contain pennies have the same size. The single small bag of quarters is twice the size of a single bag of pennies. The probability of grabbing each bag is proportional to its size.

100 people get to each randomly grab a bag from this cauldron. If I want to grab the bag of quarters, when I should I grab a bag? For instance, should I do it first? Second? Last?

My current hunch is that it does not matter. After all, the bags are randomly distributed, so this scenario should be no different from someone just randomly distributing all the bags at the same time. Is this reasoning sound?

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up vote 6 down vote accepted

If this reasoning were sound, all 100 people should have the same probability $1/100$ of getting the bag of quarters. But the probability for the first person who grabs a bag is easy to calculate; it's $2/101\ne1/100$.

The probability for the $n$-th person to get the bag of quarters is

$$ \frac{99}{101}\cdot\frac{98}{100}\cdot\dotso\cdot\frac{101-n}{103-n}\frac2{102-n}\;, $$

so the ratio of successive probablities is

$$ \frac{101-n}{102-n}\lt1\;, $$

so the probability is highest for the first person.

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+1 By the way, this problem is a good example of Wallenius non-central hypergeometric distribution. – Sasha Sep 16 '12 at 3:52

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