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This question is inspired by this question on collecting cereal toys.

Imagine a cereal manufacturer starts a promotion where their customers have to collect 5 different coupons which they can redeem for a free packet of cereal. They expect their customers to have to buy $137/12 \approx 11.4..$ packets of cereal on average to get the free packet and very roughly this is equivalent to giving a discount of 10% to their customers and they can budget for this and balance the cost against expected increase in sales. But what if large groups of friends get together and freely exchange coupons to get the 5 they require? What is the calculation?

For a single customer with no swapping the expectation is that they will have to buy on average $1 + \frac 5 4 + \frac 5 3 + \frac 5 2 + 5 \approx 11$ packets. Now if a group of 10 customers get together and swap coupons in an optimal manner, a naïve calculation might be $\left( 1 + \frac 5 4 + \frac 5 3 + \frac 5 2 + 5 \right) \frac 1{10} \approx 1 $, which is obviously not correct.

What is the correct way to do this, so that the manufacturer can correctly estimate the cost of the promotion? For the exercise assume the average size a typical group of friends is known, e.g. ten.

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    $\begingroup$ I don't know of an easy way to calculate it. Your intuition that it will take rather less than $11$ per person is correct because one person can fill in another's shortfall. If the group is huge it will take barely over $5$ coupons per person because the law of large numbers says they will be roughly evenly distributed. It depends on how they buy boxes. If they each buy $5$ and check whether they have the proper distribution it will be rare. If they each have to buy another one I would guess the chance will be high they do, so the expected value will be close to $6$. $\endgroup$
    – Ross Millikan
    Nov 20 at 14:41

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