# Low probability events and witness testimony

I experienced a low probability event yesterday.

It has according to frequencists 5% chance of occuring.

However, you know there is a 25% chance that I'm mistaken about experiencing the event at all and 70% chance that I will be lying about the event because if I get you to believe it I will earn 1 000 000 USD.

What are the % chance I'm telling the truth and the event happened?

Which strategy has the highest % chance of guessing what has really happened?

Strategy A: You trust me and the testimonies I give?

Strategy B: You ignore me and choose to believe the frequencists

• The probability that the event happened and the probability of you telling the truth are two different probabilities, both of which can be computed. Neither of your "strategies" makes sense, one uses the totality of evidence instead. The frequency is taken as the prior and Bayes updated based on your testimony. Your trustworthiness is automatically incorporated into the update, see Bayesian inference. Commented Dec 7, 2020 at 21:08
• It also highly depends on the significance of the event for me. If you tell me you saw a balcony with rare tropical flowers on it, I would say "sure, man. Why not." If you tell me you saw my daughter kissing her math teacher, although it might be more probable than tropical flowers, I would require more evidence. Commented Dec 7, 2020 at 23:06

As Conifold notes, it is not a question of choosing a strategy but of combining the prior frequency with the evidence of your testimony to get a posterior probability. This kind of thing can be done using Bayesian methods. However with your example, there is not sufficient information. You say that you might be mistaken and also that you might be lying, but we don't know how these combine. Could you be both mistaken and lying and hence accidentally telling the truth, or did you mean to rule this out? Also, over what timescale are you calling your observation low probability? We have most likely all witnessed low probability events at some time in our lives, but over a long enough period of time, this is not unusual.

Your problem is similar to one that was used by some cognitive psychologists as a test of people's ability to do simple reasoning with uncertain information. The example goes like this. A person is run over in a hit and run car accident. There is incontrovertible evidence that the car responsible is a taxi. There are two taxi companies in the town, one with green livery and the other blue. 85% of taxis are green, 15% blue. The victim is the only eye witness and claims that the taxi was blue. However, light was fading at the time and it is possible that they were mistaken. So some tests are performed in which the victim is presented with blue taxis and green taxis under similar lighting conditions and asked to identify the colour. They are found to be 80% reliable, i.e. they can correctly identify a green taxi as green and a blue taxi as blue 80% of the time. In this example, we are not being asked to consider the possibility that the witness is lying. We simply want to know, given the frequency of blue taxis and the reliability of the witness, what is the probability that the taxi was blue?

If you do the Bayesian calculation, if E is the event of the victim identifying the taxi as blue, and B is the proposition that the taxi is blue, then

``````P(B|E) = P(B).P(E|B) / ( P(B).P(E|B) + P(¬B).P(E|¬B) )

= (0.15 x 0.8) / (0.15 x 0.8 + 0.85 x 0.2)

= approx 41%
``````

The slightly surprising result is that even though the witness is 80% reliable, it is still more probable than not that the taxi was in fact green. In tests, the vast majority of people get this completely wrong. Indeed, it is not uncommon for people to ignore the prior frequency entirely and think that if the witness is 80% reliable then there must be a probability of 80% that they are correct and the taxi is blue. This is just one of many experiments that show that people are extraordinarily poor at reasoning with uncertain information. If you don't learn to use probability theory and Bayesian methods, you will almost always get examples like this wrong.

• Many Thanks Bumble! Commented Dec 8, 2020 at 8:36
• @Philosophy101 The ignoring of priors is called the base rate fallacy. Great answer, Bumble, should be accepted. Commented Dec 8, 2020 at 11:44