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A marine biologist wanted to estimate the number of fish in a large lake. He threw a net and found $30$ fish in the net. He marked all these fish and released them into the lake. The next morning he again threw the net and this time caught $40$ fish, of which two were found to be marked. The (approximate) number of fish in the lake is:

  1. $600$
  2. $1200$
  3. $68$
  4. $800$
  5. $120$
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14 votes
Answer should be $600,$ Option A.

The problem given is equivalent to the problem in which an urn contains some number of white balls in it. We take $30$ balls out of it, mark them and put them back into the urn. Now, we randomly take $40$ balls out of the urn, $2$ of them are found to be marked. What is the approximate number of balls that were present in the urn initially?

Solution: Suppose the urn contained $X$ balls initially. Then
if we take $n$ ball out of urn, probably $n\times (30/X)$ balls will be marked out of $n$ balls.
Here, $n = 40.$
So, Probably $40*(30/X)$ out of $40$ balls will be marked.

But it is given that there are $2$ marked balls,
So, $2 = 40\times (30/X),$ which gives

$X = (40 \times 30)/2 = 600.$
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Biologist, marked 30 fish

Now , Next day we get 2 fish marked among 40 fish

                             therefore 30 fish marked among 40/2 *30=600 fish
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2 votes

This is Mark and Recapture statistical method to estimate population size where it is impractical to count every individual one by one. 

Let

$N$ = Number of fishes in the population
$n$ = Number of fishes marked on the first visit $= 30$
$K$ = Number of fishes captured on the second visit $= 40$
$k$ = Number of recaptured fishes that were marked $=2$

It is assumed that all individuals have the same probability of being captured in the second sample, regardless of whether they were previously captured in the first sample. 

This implies that, in the second sample, the proportion of marked individuals that are caught $(k/K)$ should equal the proportion of the total population that is marked $(n/N)$

$$\dfrac{n}{N} = \dfrac{k}{K}$$

$$\implies N = \dfrac{nK}{k}$$

So, $N = \dfrac{30 \times 40}{2}= 600$ 

Answer:

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