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**1. Homework Statement**

There is

**k**carps in the pool,

**m**of them are marked. I randomly fish out

**n**carps and see that

**x**of them are marked.

What is the expected value of number of carps in the pool? (ie. expected value of number of the carps in the pool in the beginning)

How will the expected value change as we fish the carps out one by one? (ie. not together, but one carp at the time).

To sum it up, I know

**m**,

**n**and

**x**and I don't know

**k**.

**2. The attempt at a solution**

I'm not sure how to solve it. For the first case, I think that ratio of marked carps in the group of the carps drawn out should be (averagely) the same as the ratio or marked carps in the whole pool, ie.

[tex]

\frac{x}{n} = \frac{m}{k}

[/tex]

so

[tex]

k = \frac{m.n}{x}

[/tex]

But will it really determine the expected value of number of all carps?

For the second case, I'm even more out-of-idea.

From definition, I'd say expected value of number of the carps is

[tex]

\sum_{i=0}^{k} X_i p_i

[/tex]

Where [itex]X_i = i[/itex] and [itex]p_i[/itex] is probability that there are [itex]i[/itex] carps in total.

But this doesn't seem to be an effective approach.

Could someone give me some hint, please?

Thank you.