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Let's say I have a situation where I spend X dollars on a product that contains a set of items that go for specific individual retail prices and I know the probabilities of getting each item in the set of products I buy. Is there a way to tell if my return Y will be greater than X over the long run?

For example, say I am given a list of N (say, N=100) items along with their probabilities and individual retail prices that I buy in a set of M items, where M < N (for example, M=10), for X dollars (let's assume $10). Is there a formula I can plug these values into in order to determine if I will make profit over the long term of buying these items in random lots and selling these items individually?

I know it's easy to tell that you will make profit if all the items are worth more than \$1, and that it is easy to show that you will lose profit if all the items are worth less than \$1 individually. But how can you tell if the values and probabilities of all these items vary?

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Multiply the value of each of the possible items by the probability that that item will be in a set. Add these up. This is the "expected value" of the set. If the price of the set is less than the expected value, you can expect to turn a profit in the long run.

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