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**ronmurp**on Conjunction Fallacy · 2012-08-15T19:13:51.419Z · LW · GW

I suspect respondents are answering different questions from the ones asked. And where the question does not include probability values for the options the respondents are making up their own. It does not account for respondents arbitrarily ordering what they perceive as equal probabilities. And finally, they may be changing the component probabilities so that they are using different probability values throughout when viewing the options.

The respondents are actually reading the probabilities as independent, and assigning probabilities such as this: A: P(Accountant) = 0.1 C: P(Jazz) = 0.01 E: P(Accountant^Jazz) = P(Accountant) x P(Jazz) = 0.001, and you would expect the correct ranking

But if they are perceiving E as conditional then P(Accountant|Jazz) = P(Accountant^Jazz)/P(Jazz) = .001/.01 = 0.1, and leaving the equal ranking of A, E ordered as A, E they end up with A >= E > C. And, it's also possible they are using an intuitive conditional probability and coarsely and approximately ranking without calculation.

They may also be doing the intuitive of the following, by reading the questions in order:

A: Yeah, sounds about right for Bill. Let's say 0.1 C: Nah, no way does Bill play Jazz. Let's say zero! E: Well, I really don't think he plays jazz, and I really thought he'd be an accountant. But I guess he could be both. In this case I'm going for 0.05 accountant, but 0.02 Jazz. 0.05 x 0.02 = 0.001

So, A > E > C

In this last case the fact that he could both be an Accountant and play Jazz (E) is more plausible than he would play Jazz and not be an accountant (reading C as not being an accountant). Of course C does not rule out him also being an accountant, but that's not what appears to be the intuitive implication of C. It's as if the respondent is thinking, why would they include E if C already includes the possibility of being an accountant? And though the options are expressed as a set the respondent is not connecting them and so adapting the independent probabilities in each option. As I said, this might be quite intuitive so that the respondents do not perform the calculations and so do not see the mistake. That the question says "not mutually exclusive or exhaustive" may not register.

The diplomatic response might be explained by the following. Without any good reason respondents to (1) think suspension unlikely. Because they are not asked (2) they are asked to rate this independently of anything else, whether that be invasion of Poland, assassination of the US President, or anything else not mentioned in (1). Since they are not given *any* reason for suspension they think it very unlikely. So, your point that "there is no possibility that the first group interpreted (1) to mean 'suspension but no invasion' " does not hold. They can interpret it to mean 'suspension but nothing else'.

But in (2) the respondents are given a good reason to thank that if invasion is likely then suspension will follow hot on its heels. Also, some respondents might be answering a question such as "If invasion then suspension?", even though that is not what they are being asked.

So I think there are explanations as to why respondents don't get it that go beyond simply not knowing or remembering the conjunction condition, let alone knowing it as a 'fallacy' to avoid.

Is probability a cognitive version of an optical illusion? Two lines may not look the same length, but when you measure them they are. When two probability statements appear one way they may actually turn out to be another way if you perform the calculation. The difference in both cases is relying on intuition rather than measurement or calculation. Looked at it from this point of view probability 'illusions' are no more embarrassing than optical ones, which we still fall for even when we know the falsity of what we perceive.