Perception of randomness: On the time of streaks

People tend to think that streaks in random sequential events are rare and remarkable. When they actually encounter streaks, they tend to consider the underlying process as non-random. The present paper examines the time of pattern occurrences in sequences of Bernoulli trials, and shows that among all patterns of the same length, a streak is the most delayed pattern for its first occurrence. It is argued that when time is of essence, how often a pattern is to occur (mean time, or, frequency) and when a pattern is to first occur (waiting time) are different questions and bear different psychological relevance. The waiting time statistics may provide a quantitative measure to the psychological distance when people are expecting a probabilistic event, and such measure is consistent with both of the representativeness and availability heuristics in people's perception of randomness. We discuss some of the recent empirical findings and suggest that people's judgment and generation of random sequences may be guided by their actual experiences of the waiting time statistics.