Stochastic simulations are used to characterize the knotting distributions of random ring polymers confined in spheres of various radii. The approach is based on the use of multiple Markov chains and reweighting techniques, combined with effective strategies for simplifying the geometrical complexity of ring conformations without altering their knot type. By these means we extend previous studies and characterize in detail how the probability to form a given prime or composite knot behaves in terms of the number of ring segments, $N$, and confining radius, $R$. For $ 50 le N le 450 $ we show that the probability of forming a composite knot rises significantly with the confinement, while the occurrence probability of prime knots are, in general, non-monotonic functions of 1/R. The dependence of other geometrical indicators, such as writhe and chirality, in terms of $R$ and $N$ is also characterized. It is found that the writhe distribution broadens as the confining sphere narrows.
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