Some funny mathematical questions arise when you try to estimate the available number of balls in a jar without being able to have any look in it.
Zoologists had to solve one of them: they need to estimate the number of fishes in a lake only being able to capture some of them. A simple version of this is: "having a fishing rod, how can you estimate the number of fishes present in a lake?" (without fishing them all).
I put the answer in the comments of the post...
References
- The Estimation of Total Fish Population of a Lake search in Google.scholar
This post is part of the Carnival of mathematics.
1 comments:
Just fish for instance 200 fishes, mark them and free them back into the lake.
Few hours later, come back and fish again 200 fishes, let's say that you capture 20 marked fishes (i.e. your proportion of recaptured marked fishes is Q=20/200=10%).
You know that the estimated proportion Q of marked fishes follows a Gaussian distribution centered on the real one M and having a variance of M (1-M) / 200.
Since you observe Q=10% you can conclude that with a probability of 95%, the underlying proportion of fishes is between
Q-1.96*sqrt(M(1-M)/200)
and
Q+1.96*sqrt(M(1-M)/200).
In this case between 6% and 14%.
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