Let D denote the fundamental discriminant of a real quadratic field, and let h(D) denote its associated class number. If p is prime, then the ’Cohen and Lenstra Heuristics‘ give a probability that p[nmid]h(D). If p>3 is prime, then subject to a mild condition, we show that $\# \{0<D<X|p\nmid h(D)\}\gg_p \frac{\sqrt{X}}{\log X}.$ This condition holds for each 3<p<5000.