This paper considers linear fair risk sharing rules and the conditional mean risk sharing rule for independent but heterogeneous losses that are gathered in an insurance pool. It studies the asymptotic behavior of individual contributions to total losses when the number of participants to the pool tends to infinity. It is shown that (i) insurance at pure premium is obtained for an infinitely large pool and (ii) the difference between the actual contribution and the pure premium becomes ultimately Normally distributed. The linear fair risk sharing rule approximating the conditional mean risk sharing rule is then identified, providing practitioners with a useful simplification applicable within large pools. Also, the approximate number of participants required to keep the volatility of individual contributions within an acceptable range is obtained from the established asymptotic Normality.