Probability of determining the total number of events from incomplete data?

Question

Mathematics

The task originated in a cluster serving client requests.
Given:
N empty baskets. A blindfolded person throws one apple into the baskets, the probability of hitting any
of the baskets is 1\N (misses are not possible). In total, M apples were thrown, but no one counted them during the throws.
Then the person's eyes are untied and one basket is brought (any one to choose from).
Question: what should be the number L of apples in this one basket, so that a person, knowing about the probability of hitting 1 \ N, could name the number M of thrown apples with a probability P > 99.9%?

Answer 1

Endless. Then we can be guaranteed to say that an infinite number of apples have been thrown. In all other cases, we can only say that M ≥ L.

Answer 2

Well, in the case of one basket, you can calculate exactly, with a probability of 1. And in the case of an infinite n, the brought basket will be rather empty and the probability of a correct answer is 0. Between these values, you can probably build a function, a probability distribution, with arguments. I'm not sure further...
(edition)
Considering that I always answer that M=L*N, then L can not be used further. Anything below that starts with >, substate at www.wolframalpha.com/input, for verification and clarification. And it turns out that even with a small N = 2 (we have already considered = 1), the best that can be achieved is ~0.5, and if N = 3, then everything is even worse.
> Plot3d[(M choose M/N)/N^M, {N, 1, 10}, {M, 0, 100}]
M=2
N=2
> (2 choose (2/2))/(2 ^2) = 0.5
M = 100
N = 2
> (100 choose (100/2))/(2^100) ~ 0.08
M = 1000
N = 2
> (1000 choose (1000/2))/(2^1000) ~ 0.025
M = 100000
N = 2
> ( 100000 choose (100000/2))/(2^100000) ~ 0.025....
M = 9
N = 3
> (9 choose (9/3))/(3^9) ~ 0.004
M = 18
N = 3
> (18 choose (18/3))/(3^18) ~ 0.00004

Answer 3

Take the binomial distribution formula and test the hypothesis that the observed value of L corresponds to the mode of the binomial distribution for M, against the (possibly non-modal values) for (M-1) and (M+1). By increasing L, make sure that the hypothesis is not rejected with the ALPHA level you require.
Perhaps, looking at the nature of the binomial distribution for large L, you will not succeed. That is, it is impossible to say with an accuracy of M at the level you want - only the range - from M_LO to M_HI