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Suppose, somebody has to estimate the likelihood of one of the following events (or has to estimate which event is more likely):

  1. A coin is tossed six times and each time the result is heads. (combined)
  2. A 64-sided die is rolled and the result is 1. (uncombined)

Since 2⁶ = 64, both events are obviously equiprobable. However, somebody with less statistical prowess might think otherwise. Also, in a more complex scenario (e.g., in a board game) at least one of the probabilities may be impossible to calculate for almost anybody (at least in a short time).

I am interested in studies that investigate, whether combined probabilities (1) are estimated to be higher, lower or equal than uncombined probabilities (2), even if the actual probabilities are identical.

I am also interested in studies, which investigate the influence of the presentation of the scenario on the estimation of probabilities: Is the same scenario perceived differently, if presented in a way that emphasises combinedness? For example, one could simply ask somebody to estimate the probability that one of his close relatives will die within a year (uncombined) or one could ask him to list all his close relatives, their age and state of health first (combined). After all, a lot of probabilities with real-life applications are combined.

Note that I am not interested in which of the estimates is more accurate, only which one is higher.

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Welcome to CogSci. This sounds like an interesting question, but after the example in the last paragraph, I'm not sure I understand what you mean by cumulative and non cumulative. The term cumulative as I know it refers to questions like "what is the probability of at least 4 heads in tosses", while non cumulative would be "what is the probability of exactly 4 heads in 6 tosses". Is that what you mean? If not, can you be more specific and define what you mean by the term? –  Ofri Raviv May 4 '13 at 17:28
    
Thanks, I forgot that cumulated is a predefined term in statistics and I changed it to combined, which should be the correct term. I denote an event as combined if it occurs if and only if a collection of other events occur. –  Wrzlprmft May 4 '13 at 18:20
    
A related issue is the well known phenomenon (I believe I read it in Kahneman/Tversky but it might be earlier) where adding a plausible-seeming extra condition to a combined probability makes people to perceive as more likely despite being less likely. The classic example was "USA breaking diplomatic relations with USSR" perceived by experts as less likely than "USSR invading Poland and USA breaking diplomatic relations with USSR" - but the second is clearly a subset of the first one. –  Peteris Feb 24 at 17:25
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2 Answers 2

up vote 5 down vote accepted

The probability of conjunctive events (all six tosses are heads) are overestimated, relative to a single event of similar overall probability.

This result has been shown by Paul Slovic, in an experiment that is described in its abstract as follows:

This study examined the effects on the attractiveness of a gamble, of manipulating the number and structure of the independent events on which the gamble's outcomes were contingent, while holding the expected value constant. 3 manipulations were studied. The most effective was the situation where a gamble offering a payoff with probability equal to p was changed into a gamble whose payoff was contingent on the joint occurrence of 4 independent events, each having a probability equal to p^(1/4). Although individual differences were substantial, the majority of Ss behaved as if the probability of the compound event was much greater than p, its true value.

Similar results were obtained by Maya Bar-Hillel, in several experiments. In one of them she presented subjects with displays similar to this one:

Fig. 1 from Bar-Hillel, 1973. An example of an array in the experiment

A path in this array is defined as any line which originates at any element of the first row, connects it with any element in the second row, proceeds to pass through any element in the third row, and so on to the last row.

The subjects were asked to estimate the proportion of paths in such displays that cross only X's, relative to all paths. Results show a robust (and quite large) overestimation of this proportion.

References

  • Slovic, P. (1969). Manipulating the attractiveness of a gamble without changing its expected value. Journal of Experimental Psychology, 79(1p1), 139.

  • Bar-Hillel, M. (1973). On the subjective probability of compound events. Organizational behavior and human performance, 9(3), 396-406.

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I believe these questions are dealt with by "support theory," the seminal publications being:

  • Tversky, A., & Koehler, D. J. (1994). Support theory: A nonextensional representation of subjective probability. Psychological Review, 101(4), 547-566.
  • Rottenstreich, Y., & Tversky, A. (1997). Unpacking, repacking, and anchoring: advances in support theory. Psychological review, 104(2), 406.

One of the basic findings accounted for by support theory is an unpacking effect whereby people judge an event that implicitly consists of a disjunction of many events (e.g., the probability that a person will "die from a natural cause") as less probable than the sum of the probability estimates for some component events that one could "unpack" the original hypothesis into (e.g., the sum of [a] the probability a person will die from heart disease, [b] the probability a person will die from cancer, [c] the probability a person will die from other natural causes). For example, in one study reported in Tversky & Koehler (1994), participants on average judged the probability of the implicit disjunction above ("death from a natural cause") to be 58%, and the probability of the unpacked version to be 22% + 18% + 33% = 73%. In the words of Tversky & Koehler, "unpacking an implicit disjunction may increase, but not decrease, its judged probability" (p. 549).

There is a lot more to support theory than just this unpacking effect, but a more complete summary of the theory is beyond my knowledge.

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Thank you. These publications are certainly very helpful, especially as a starting point for further literature survey. Unfortunately, support theory and all I found so far only seems to deal with the case of additive probabilities (as in the example you reported) and not with multiple probabilities. –  Wrzlprmft May 5 '13 at 13:03
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