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The experiment was designed to test the hypothesis, derived from earlier investigations, that people tend to overestimate compound probabilities, in the rough sense that they think they have a better chance of success than is actually the case. Fifty boys, aged 14+, were presented with various m × n arrays, where m represents the number of stages and n represents the number of alternatives at each stage. There was one correct but unknown alternative at each stage and, in order to win a prize, the subject had to guess them all in turn, only one guess being allowed at each stage. The task of the subject was then to equate what he believed was his chance of winning the prize with one of a set of lotteries. The chance of success in these lotteries ranged from 0·9 to 0·0001. The hypothesis was confirmed and, in addition, what seems to be a ‘law’ of sequential choice (or decision) seemed to emerge. The ‘law’ is such that if the number of stages in the array is held constant, the relative overestimation of the chance of guessing correctly at all stages varies directly with a power of the number of alternatives per stage. If, however, the number of alternatives is held constant, the relative overestimation varies exponentially with the number of stages. This ‘law’, which we call ‘the inertial-ψ effect’, appears in almost identical form in two independent experiments. It may prove to have a very general application in the characterization of human judgement and its fallibility in many spheres of life.