To determine a number of trout in a small lake you proceed a

To determine a number of trout in a small lake, you proceed as follows: catch 100 trout, tag each one of them, and put them back into the lake. A month later you catch again 100 trout and notice that 10 of them are tagged. a) Give a rough estimate of the number of trout in the lake. b) Let N be the number of trout in the lake. Find an expression, in terms of N, for the probability that you would catch 10 tagged trout out of the 100 you caught the second time.

Solution

a)

This would mean that you have tagged 10/100 or 10% of the trouts.

Thus, you estimate of the number of trouts is

N = n/p = 100/0.10

N = 1000 [ANSWER]

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b)

If N = the population size, then there are NC100 ways to choose any 100 trouts.

There are 100C10 ways to choose 10 marked, and (N-100)C90 ways to choose 90 untagged.

Thus,

P(10 tagged) = (100C10) [(N-100)C90] / (NC100)

Where,

NCr = N!/(r! (N-r)!) [ANSWER]