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I am aware that $P(A\cap B)$ is less than $P(B)$. I have already seen Probability Question: Would A always have a greater chance of $A\cap B$?.

My issue is regarding the conjunction fallacy and order of events. Below is a common example of the fallacy, where logically $2$ should be more likely than $1$ as $P(A\cap B)$ is less than $P(B)$:

1. The Soviet Union will attack Poland and then the USA will cut off diplomatic relations with it.

2. The USA will cut off diplomatic relations with the Soviet Union.

Now assume A is the attack event and B is the event cutting off diplomatic relations.

The problem here is that looking at $P(A\cap B)$ in this way doesn't seem to take order of events into account. Won't the posterior probability $P(B|A)$ (in situation $1$) increase dramatically as compared to the prior $P(B)$ (situation $2$)?

Now if the probability of an attack, $P(A)$ is high as well (let's assume it is for mathematical purposes) this should be enough to make $P(A)\times P(B|A)$ more likely than $P(B)$ alone.

But this argument justifies the fallacy. So where is the logical error, or is this example invalid?

Stormlight
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1 Answers1

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Indeed, the posterior probability $P(B\mid A)$ could be significantly higher than $P(B)$. We have

$$ P(B\mid A)=\frac{P(A\cap B)}{P(A)} $$

and

$$ P(A\mid B)=\frac{P(A\cap B)}{P(B)} $$

and thus

$$ \frac{P(B\mid A)}{P(B)}=\frac{P(A\mid B)}{P(A)}\;. $$

Thus, $A$ makes $B$ more likely by the same ratio as $B$ makes $A$ more likely. If telling you that diplomatic relations were cut off would make you find it twice as likely that Poland was attacked, then telling you that Poland was attacked should make you find it twice as likely that diplomatic relations were cut off, and vice versa.

And here's what prevents the paradox you're worried about: If $P(A)$ is already very high, then this ratio can't be very high. If you already think that an attack is very likely, telling you that diplomatic relations were cut off can't make it much more likely. By the above, then, telling you about an attack also can't make it much more likely that diplomatic relations were cut off. Since you already thought that an attack was highly likely, telling you that an attack did actually occur doesn't give you much information that you didn't already have, so it can't change your beliefs about the likelihood of other events very much.

The order of events has nothing to do with all this. Conditional probabilities don't deal with causality. The fact that $A$ is more likely to cause $B$ than vice versa doesn't introduce any asymmetry into the calculus of probabilities for these events. Imagine looking at a movie of the events shown in reverse. The fact that in the movie cutting off relations “causes” the attack doesn't change anything about the probabilities you assign to seeing various conjunctions of events in the movie.

joriki
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