Before revision:  Our results provide a mathematical and statistical foundation for the design and validation of experiments that directly compare deviations from randomness with human beliefs about deviations from randomness in Bernoulli sequences .
After revision:  Our results provide a mathematical and statistical foundation for the design and validation of experiments that directly compare deviations from randomness with human beliefs about deviations from randomness , and thereby constitute a direst test of the hot hand fallacy .