  In order to study the galaxy population of galaxy clusters with photometric
data one must be able to accurately discriminate between cluster members and
non-members. The redMaPPer cluster finding algorithm treats this problem
probabilistically. Here, we utilize SDSS and GAMA spectroscopic membership
rates to validate the redMaPPer membership probability estimates for clusters
with $z\in[0.1,0.3]$. We find small - but correctable - biases, sourced by
three different systematics. The first two were expected a priori, namely blue
cluster galaxies and correlated structure along the line of sight. The third
systematic is new: the redMaPPer template fitting exhibits a non-trivial
dependence on photometric noise, which biases the original redMaPPer
probabilities when utilizing noisy data. After correcting for these effects, we
find exquisite agreement ($\approx 1\%$) between the photometric probability
estimates and the spectroscopic membership rates, demonstrating that we can
robustly recover cluster membership estimates from photometric data alone. As a
byproduct of our analysis we find that on average unavoidable projection
effects from correlated structure contribute $\approx 6\%$ of the richness of a
redMaPPer galaxy cluster. This work also marks the second public release of the
SDSS redMaPPer cluster catalog.
