Mathematically, the fallacy results from misunderstanding the concept of a conditional probability, which is defined as the probability that an event A occurs given that event B is known – or assumed – to have occurred, and it is written as P(A|B).

The error is based on assuming that P(A|B) = P(B|A).

For example, let A represent the event of getting a haircut, and B the event of reading a haircut meme.

But this equality is not true: in fact, although P(A|B) is usually very small, P(B|A) may still be much higher.

[–] roastpotatothief@lemmy.ml 1 points 3 years ago* (last edited 3 years ago) (1 children)

No I don't think that's it. I would express it this way.

The chance of me reading a haircut meme straight after getting a haircut is near zero. But there are multiple people reading the haircut meme. So if you post a haircut meme, there is a high chance that ANY ONE of the readers will be straight after getting a haircut.

Like Sally Clark was convicted because the chance of having two miscarriages in a row is a million to one - it's not plausible. The only other explanation is that she murdered her babies, so that's the only plausable explanation.

But in a population of 60 million people, you are likely to find someone (or 60 people) who have had two miscarriages in a row. She was just the unlucky one.

Wikipedia has more detail and intersting angles, like the defendent's fallacy, why it is a fallacy and when it might be correct.

Then there's more stuff, and what happened after the conviction... That pathologist was never even charged with a crime, or lynched.

edit:

But your point is interesting too. The chance of winning the lottery given that I've just had a coffee is very low. But the chance that I've just had a coffee given that I've won the lottery is very high. But if I drink a coffee and then immediately win the lottery, people might assume that having just had a coffee improves my lottery chances.

I don't know if there is a name for that one, but it sounds like something people do a lot.

I get sick, take medicine, get better. Then chance of having taken medicine given that I get better is 100%. So people will assume the medicine helps. But the chance that I get better given that I've taken medicine might be 10%. The chance I get better if I had not taken medicine might be more than 10%, but that result is not obvious at all.

[–] tomtom@lemmy.ml 1 points 3 years ago

Ah your explanation clears it up. That whole conditional probability thing is in the wikipedia article, but I see now that my explanation of the haircut thing was not correct.

I guess maybe this is a better formulation:

p1 = P(not being guilty | evidence found)

p2 = P(evidence found)

Prosecutor's fallacy would assert that, if p2 is small say 0.01%, then the defendent is guilty. But really the relevant probability is p1, which could be quite a bit larger than 0.01%.

Anyways let me know if you agree lol.