Do syllogisms apply to probabilistic statements? I came across a statement that triggered my logic 101 alarm on another SE site, regarding risk factors for an illness. The statement was similar to this (modified): > Favoring green jelly-beans is one of the risk factors for paralytic dispepsia, i.e. paralytic dispeptics are more likely to favor green Jelly-Beans. I parsed this sentence as: Favoring green jelly beans increases the probability that you'll suffer tummy aches. _Therefore:_ If you have tummy aches, you are more likely someone who favors green jelly-beans. I know That "A implies B" is not equivalent to the proposition "B implies A" (but rather, equivalent to "Not(B) implies Not(A)") so I'd dismiss that example as a fallacy _if_ there was not an element of probability involved. But given it _is_ a probabilistic statement I'm unsure if this is valid inference or not. Am I wrong in applying that syllogism in this case?

We are not in a logical setting here, we are not talking about implication, we are talking about conditional probability. Let's take this example


p(i prefer green jelly beans) = 0.5
p(i have tummy aches | i pefer green jelly beans) = 1
p(i have tummy aches | i do not prefer green jelly beans) = 0.5


These conditional probabilities contain way more information than just 'i prefer green jelly beans' => 'i have tummy aches'. If i have tummy aches, then the likelihood of me prefering green jelly beans is `0.5*1 + 0.5*0.5 = 0.75`!

So yes, it is correct.