Probability learning in 1000 trials.

This paper reports a simple two-alternative noncontingent probability learning experiment with an uncon-ventional feature: each 5 made 1000 consecutive predictions, making possible very detailed analysis of responses which occurred after learning was essentially completed. Some abbreviations will be useful. The 5 1 predicts either L or R; after each prediction he observes either 1 or r. The probability that 5 will make prediction L on trial t + 1 will be called p t. The probability of 1 on any trial is a constant for any given 5; it will be called-IT. The occurrence of a prediction will be called a response ; the occurrence of a display of an event following a response will be called an outcome. An outcome follows each response; the nature of the outcome is independent of the nature of the response. The interpretation of this experiment will focus on three issues: 1. The probability matching hypothesis. The probability matching hypothesis (PMH) asserts that the asymptotic probability of choice, p x (p x = lim£<; i—»-oo it is assumed that this limit exists) by the equal-alpha case of the Bush-Mosteller learning model (Bush & Hos-teller, 1955), and has been supported by a number of experiments, though not by others. 2. The extreme-asymptote generalization. In 1956 I reported an experiment which argued against PMH and in favor of a theory about /> " which I call the extreme-asymptote generalization. That generalization asserts that p x is more extreme than T, and as the absolute value of the difference between TT and 0.5 increases the difference between p x and TT also increases until p x becomes 1 or 0. As stated, this hypothesis makes only ordinal predictions; a way of making it yield ratio scale predictions (and also of applying it to situations in which amount of payoff is varied) is discussed in Edwards (1956) and applied later in this paper. 3. Sequential dependencies, the gamb-bler's fallacy, and path independence. Stochastic learning theories often assume that the effects of events prior to a given trial are summarized in a set of probabilities for the responses available on that trial; this assumption is known as the path independence assumption (for a better definition, see Bush & Hosteller, 1955, p. 17). Contradictory to this is the common observation that if a flipped coin comes up heads eight or nine times in a row, 5 is likely to decide …