# Estimation population size using sampling - real life probability

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?

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?"

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.