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I have come across similar questions like this previously but I just can't get my head around a correct method

Clive catches 50 bees from the beehive and marks each bee with a dye then lets them go.

The next day he catches 40 bees from the hive. 8 of these bees have been marked with dye.

Work out an estimate for the number of bees in the beehive

I has a slight guess that the probability of picking 8 out of the 40 in the second day is the same as the probability of picking 50 out of the total $N$ in the first day so $\frac{8}{40}=\frac{50}{N}$ to give $N=250$ which I think was correct, but I am struggling to convince myself that the logic behind this is true

Also I tried another method using probability but it didn't work and I really don't know why. I used the fact that $ P(B|A)=\frac{P(A and B)}{P(A)}$ so if $N$ was the total population that we tried to find, then the $P(A and B)$ = $\frac{82}{N}$ , $P(A) =\frac{50}{N}$ and the probability of B given A is the 8 out of the total bees so $P(B|A)= \frac{8}{N}$ Hence from this we work out that $N=4.878$ which really cannot be true

Please could someone unpick where I have went wrong and perhaps suggest a better way of tackling these types of animal sampling questions?

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According to the Maximum Likelihood estimation your estimation should be $250$ bees.

By seeing that $8$ out of the $40$ bees you caught the second time, are marked, you assume that the $\frac{8}{40}=\frac{1}{5}$ of the total population of the bees are marked, so you say that if the $\frac{1}{5th}$ of the population is 50, then the population must be $5*50=250$.

But why do we assume that $\frac{8}{40}=\frac{1}{5}$ are marked?

This is the Maximum Likelihood estimation.

Let's define $\theta$ as the percentage of marked bees after your first visit in the hive.

Let's also define $X$ as the number of marked bees you found on your second visit in the hive and $n$ the total number of bees you found on your second visit in the hive.

Then the function:

$L(\theta)=\binom{n}{X}\theta^X(1-\theta)^{n-X}$

gives you the probability of getting $X$ marked bees out of $n$ total bees for a percentage $\theta$ of marked bees in the population. If you maximize this function you will find out that it has a maximum for $\theta = \frac{X}{n}$.

Here the function is:

$L(\theta)=\binom{40}{8}\theta^8(1-\theta)^{40-8}$

and it is maximized for $\theta=\frac{1}{5}$.

Generally speaking the method of Maximum Likelihood estimate is like asking yourself:

"From which population did this data came?"

And answering:

"From the population where this data would be most likely to appear"

Here the "population" of the answer would be a population with $\frac{1}{5}$ marked bees. So the 50 (marked) bees must be the $0.2$ of the total number of bees, thus making the the total number of bees 250 bees.

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  • $\begingroup$ I understand your explanation now, thankyou. So would it be wrong to calculate it using probability i.e with my second method? $\endgroup$
    – yt.
    Jun 8 '19 at 8:09
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    $\begingroup$ @yt. If I understand correctly you are trying to use the Bayes' theorem. This theorem has no power here because it is used to calculate conditional probability. Here your goal is to make an estimation about a number (the total number of bees), not to calculate the conditional probability of an event (that is the probability of A ocuring given that B happened) $\endgroup$ Jun 8 '19 at 12:03

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