Intuitively, how many balls are in the bag?

Question

You have a bag that contains all white balls, though you have no idea how many. You reach into the bag and pull out 10 white balls. Then, with the help of a red marker, you mark those 10 white balls with a big "X" , and then put them back into the same bag. You shake up the bag, and then pull out 10 balls again. Four of the 10 balls are marked with "X". Intuitively, how many balls are there in the bag?

My answer:

Since we got 4 marked balls, there should be at least 6 unmarked balls in the bag. So, there should be at least (10 marked + 6 unmarked=) 16 balls in the bag.

Is my approach correct? Or is there any other way to find the answer?

Answer 1

$16$ is a minimum bound for the reasons you stated, but different methods will give you different numbers of balls.

I'd say that there's $25$ balls, because on the second draw, $40\%$ of the balls you drew were marked. There are $10$ marked balls in total, so assuming that $40\%$ of the entire bag is marked, given the number of total balls $n$,

$$0.4n=10$$

so $n=25$. Of course it might actually be $24$, or $26$, or any other $n>16$, but the probability of $n$ being very far from $25$ is low. You can get a range of balls using confidence intervals, but I won't bother with that here.

Answer 2

You should expect that $40\%$ of the balls are the ones you marked, because $40\%$ of the balls you drew were marked. That is, $.4(N)=10$, where $N$ is the number of balls in the bag. Solving, we find that $N=25$.

This is how many balls you might expect there to be, of course this is just one trial, so it doesn't really tell you that much.