  In a fundamental test of quantum mechanics, we have observed 228~000 quantum
jumps of a single trapped and laser cooled $^{88}$Sr$^+$ ion. This represents a
statistical increase of two orders of magnitude over previous similar analyses
of quantum jumps. Compared to other searches for non-randomness in quantum
mechanical processes, using quantum jumps simplifies the interpretation of data
by eliminated multi-particle effects and providing near-unit detection
efficiency of transitions. We measure the fractional reduction in the entropy
of information to be $< 6.5~10^{-4}$ when the value of any interval between
quantum jumps is known. We also find that the number of runs of successively
increasing or decreasing interval times agrees with the theoretically expected
values. Furthermore, we analyze 238~000 quantum jumps from two simultaneously
confined ions and find that the number of apparently coincidental transitions
is as expected. Finally, we observe 8400 spontaneous decays of two
simultaneously trapped ions and find that the number of apparently coincidental
decays from the metastable state agrees with the expected value. We find no
evidence for short- or long-term correlations in the intervals of the quantum
jumps or in the decay of the quantum states, in agreement with quantum theory.
