Q: Probability of min count in random selection Given the following scenario:


*

*5 items out of 50 are selected randomly without repetition

*The experiment is repeated 1000 times (Bernoulli)


What is the probability that any item gets selected at least 30 times in total throughout all 1000 experiment runs?
Is this formula somehow getting close?
$$
\sum _{i=0}^5 \left(\sum _{j=30}^{1000} \binom{1000}{j} \left(\frac{1}{50-i}\right)^j \left(1-\frac{1}{50-i}\right)^{1000-j}\right)
$$
