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Can Bayes's formula be applied usefully to controversial public arguments? For example (and here I'm attempting to choose a real example but also avoid a political discussion), if 900 out of 1000 people believe the OJ Simpson murdered Nicole Brown Simpson and 100 out of 1000 people believe he didn't, does this provide any useful information about the likelihood that Simpson murdered Nicole Brown?

This might be what the original question-poster was attempting to answer with his blue car question, using a cleaned-up example.

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> if 900 out of 1000 people believe the OJ Simpson murdered Nicole Brown Simpson and 100 out of 1000 people believe he didn't, does this provide any useful information about the likelihood that Simpson murdered Nicole Brown?

Not in the same way. Note the key assumption in the accepted answer: a 10% false positive rate. That is, we assume (for good reason) that on average the population is fairly accurate at identifying and naming colors correctly.

The analogous assumption in the OJ example would be "given media-filtered information about an emotionally-charged murder trial, most people accurately assess guilt with 90% probability." This is clearly false.

But note that our entire criminal justice system does assume that "given all the facts as presented by a prosecutor and defense attorney over the course of a trial, people instructed to vote 'not guilty' unless they are sure of guilt 'beyond a reasonable doubt' will have a very low false positive rate." And here the exponent is only 12.

That was not a key assumption. Changing it to a 49% false positive rate did not affect the result substantially.

Well, ok, fair enough, but I was using that number as an example. The point is you can look at the term (false positive)^900 and see that its tininess will dominate.

Bayes' formula can be applied to compute probabilities for basically anything, but you have to watch your assumptions. For example, the example crucially depends on every person actually having seen the car, and their answers being conditionally independent given the color of the car. If e.g. only 10 had seen the car and each told 100 others, that works out to a completely different number.

If you want to make a probabilistic argument in a public debate, you probably won't have enough information to reach reliable numbers, and it won't convince anyone who doesn't already agree with your conclusion. (Assuming they didn't doze off when you mentioned math ...)

This is an interesting point, although I think the cleaned-up question misses the mark. Most people asked about that murder would have similar definitions of 'murder', and you'd basically be gathering predictions about the state of the world.

Many people asked about a blue car will have different standards for 'blue', so our data is distorted by the possibility that people can agree on the hue of the car, but not the 'blueness' of it.

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