A (usually very good) approximation is based on the procedure called poissonization.
Consider that you have a random number of friends, with a Poisson distribution of parameter $F=5000$ and that there are $J=365$ days in a year. Then each day of the year receives a random number of friends, with a Poisson distribution of parameter $F/J$ and (this is the crucial feature) these are independent.
Then the probability $P$ that at least one day stays free is $1-p^J$ where $p$ is the probability that a Poisson random variable with parameter $F/J$ is at least $1$. One gets $p=1-e^{-F/J}$ and $ P=1-(1-e^{-F/J})^J. $ Numerically, this yields $P=.0410170619 \%$.
The quality of the approximation (which may be quantified) is based on two facts. First, the Poisson model conditionally on the total number of friends coincides with the original model. Second, at least when the parameter $F$ is large, the Poisson random variable is highly concentrated around its mean $F$, hence the conditioning is in fact not necessary.