Probability of fish math problem

[Shockwave]

A woman wants to estimate the number of fish remaining in a lake after an oil spill. She catches 50 fish and marks them. Later on, she again catches 50 fish and discovers that 10 of them are marked.

a. What is the probability of this later event if the lake contains n fish?

b. How can such data be used to estimate the number of fish remaining in the lake?

So suppose an isolated area has n creatures that we want to investigate.

She marked 50 fishes. Later she caught 50 and found 10 are tagged.

The probability of this happening is

(50 chooses 10)(n-50 chooses 40)/(n chooses 50)

Let's called the above probability f(n).

Of course, if there were n - 1 fishes, then f(n - 1) < f(n).

Using the inequality f(n - 1) < f(n), we should be able to solve for n.

Am I correct?

Thanks

 

[Tide]

Your f(n) is correct though your inequality doesn't make sense.

In your problem, f(n) will have a maximum at n = 250 (which you would expect from the simple proportion 10/50 = 50/n). I.e. 250 is the most probable number of fish in the lake so the probability of more or fewer will be less. You can obtain the 250 by plotting f(n) and is a useful exercise because it gives you an idea of how confident you can be in asserting there are 250 fish in the lake.

 

[Shockwave]

So, (50 chooses 10)(n-50 chooses 40)/(n chooses 50) reduces to 10/50 = 50/n? If so how did you reduce it please?

Thanks

 

[Tide]

So, (50 chooses 10)(n-50 chooses 40)/(n chooses 50) reduces to 10/50 = 50/n? If so how did you reduce it please?

Thanks

No, it does not! The expression you have gives the probability of pulling 50 fish with 10 of them marked from the lake if the lake contains n fish given that 50 fish were marked to begin with. I suggested graphing the function f(n) which will reveal to you that the most probable number of fish in the lake is 250.

Your second question is not the same. You were asked "how can such data be used to determine the number of fish remaining in the lake?" This is an easier question. The woman tagged 50 fish. The sample she drew from the lake contained 50 fish of which 10 were tagged meaning that 1 in 5 of the sample are tagged fish. If that ratio holds true for the entire population then 10/50 = 1/5 = 50/n so that n = 250 which agrees with your expectation from the first part!