Subjective Probability: A Judgment of Representativeness

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ii. Reflection of Randomness

Irregularity and local representativeness affect judgments of randomness. Things that do not appear to have any logical sequence are regarded as representative of randomness and thus more likely to occur. For example, THTHTH as a series of coin tosses would not be considered representative of randomly generated coin tosses as it is too well ordered.

Local representativeness is an assumption wherein people rely on the law of small numbers, whereby small samples are perceived to represent their population to the same extent as large samples. A small sample which appears randomly distributed would reinforce the belief, under the assumption of local representativeness, that the population is randomly distributed. Conversely, a small sample with a skewed distribution would weaken this belief. If a coin toss is repeated several times and the majority of the results consist of "heads", the assumption of local representativeness will cause the observer to believe the coin is biased toward "heads".

A major characteristic of apparent randomness is the absence of systematic patterns. For example- A sequence of coin tosses, which contains an obvious regularity is not representative. Thus, alternating sequences of heads and tails, such as HTHTHTHT or TTHHTTHH, fail to reflect the randomness of the process. Indeed, Sample size judge such sequences as relatively unlikely and avoid them in producing simulated random sequences. Some irregularity is expected, not only in the order of outcomes, but also in their distribution.

Sampling Distribution

They have proposed that sample size assign probabilities to events so that the more representative events are assigned higher probabilities, and equally representative events are assigned equal probabilities. The implication of this hypothesis for the study of subjective sampling distributions, i.e., the probabilities that Sample size assign to samples of a given size from a specified population. When the sample is described in terms of a single statistic, e.g., proportion or mean, the degree to which it represents the population is determined by the similarity of that statistic to the corresponding parameter of the population. Since the size of the sample does not reflect any property of the parent population, it does not affect representativeness. Thus, the event of finding more than 600 boys in a sample of 1000 babies, for example, is as representative as the event of finding more than 60 boys in a sample of 100 babies. The two events, therefore, would be judged equally probable, although the latter, in fact, is vastly more likely.

Posterior Probability

In the sampling distribution section, Sample size estimated the probability that a certain sample had been drawn from a given population. A related task, which has become increasingly popular in recent years under the impact of the Bayesian approach to statistical inference, is the evaluation of posterior probability. In this Sample size estimate the probability, or the odds, that a given sample has been drawn from one rather than another population. A typical experiment is run as follows: the S is shown two book bags, one containing, say, 80% red poker chips and 20% blue poker chips, and the other containing reversed proportions of red and blue chips. One of the bags is selected by chance and a random sample is drawn from it. The S observes the number of red and blue chips in the sample, and estimates the posterior probability, or the odds, that the sample has been drawn from the predominantly red bag. Alternatively, the S may be required to evaluate the posterior probability that a sample of observations, say measurements of height, has been drawn from a population of men rather than of women. To obtain the correct answer, one computes the probabilities of obtaining the observed sample under each of the two hypotheses. The objectives of posterior odds in such problems are simply the ratio of these two probabilities, known as the likelihood ratio. Posterior probabilities have been investigated in two different contexts: the revision of opinion and the evaluation of evidence. Revision of opinion is studied with a sequential procedure in which S is presented with successive data and revises his posterior estimate after each datum. Evaluation of evidence is studied with an aggregate procedure in which S is presented with the entire data at once and makes a single estimate of posterior probability. The most obvious fact about the sequential procedure is that Ss generally revise their opinions in the correct direction after each datum: a red chip increases their confidence that the sample has been drawn from the predominantly red bag. Consequently, the subjective posterior estimates appear monotonically related to the objective posterior probabilities. The subjective odds, however, are generally conservative, i.e., too close to unity. G These findings have fostered an approach which adopts the normative Bayesian rule as a basic model of the behaviour of the S. Man is viewed as a conservative Bayesian estimator, and his deviations from the norm are attributed to misperception of the impact of each datum, disaggregation of the joint impact of data, or to a response bias against extreme estimates. All three hypotheses predict that, within any experimental situation, subjective posterior estimates should be monotonically related to the correct Bayesian values. According to this approach, therefore, Sample size estimates are assumed to be qualitatively compatible with the normative model

Conclusion

The results and speculations presented in this paper provide merely an outline of a heuristic approach to the study of man’s competence and performance as a judge of uncertainty. The data base is admittedly narrow, and the more interesting problems of evaluating uncertainty in everyday life have yet to be faced. Nevertheless, it is evident that the present approach differs markedly from the normative approach in that it focuses on the question “how do people evaluate uncertainty?” rather than on “how well do people evaluate uncertainty?”

The decisions we make, the conclusions we reach, and the explanations we offer are usually based on our judgments of the likelihood of uncertain events such as success in a new job, the outcome of an election, or the state of the market. . An extensive literature have been devoted to evaluate how people perceive, process and evaluate the probabilities of uncertain events in context of probability learning, intuitive statistics, and decision making under risk. Most general conclusion obtained from numerous findings

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