We report on six experiments studying participants’ predictions of the next outcome in a sequence of binary events. Participants faced one of three mechanisms generating 18 sequences of 8 events: a random mechanical bingo cage, an intentional goal-directed actor, and a financial market. We systematically manipulated participants’ beliefs about the base rate probabilities at which different types of outcomes were generated by each mechanism. Participants either faced unknown (ambiguous) base rates, a specified distribution of three equiprobable base rates, or a precise, stationary base rate. Six target sequences ended in streaks of between two and seven identical outcomes. We focused on participants’ predictions of the ninth, unobserved outcome in each of these target sequences. Across all generating mechanisms and prior belief conditions, the most common prediction pattern was best described as close-to-rational belief updating, producing an increasingly strong bias toward repetition of streaks. The exception to this generalization was for sequences generated by a random mechanical bingo cage with a precise, stationary base rate of .50. Under these conditions, participants exhibited a bias toward reversal of streaks. This effect was irrational, given our instructions on the nature of the generator. We conclude that the dominant judgment habit when predicting outcomes of sequences of binary events is reasonable belief updating. We review alternate accounts for the anomalous judgments of sequences produced by random mechanical devices with a precise, stationary base rate.