Sorry, to give an answer anchored less in math or philosophy than in cognitive behavioral therapy. But in this case, it is the only relevant model that makes any sense in my opinion.

First of all, subjective measures of probability are generally horrible, and Person A's are likely to be absolute nonsense. So this form of argument in terms of absolute probability is seldom meaningful. But let's put that aside and assume this is not just arrogance.

Second, even a totally statistical application of the concept of risk is in terms of expected value, not mere probability. So you are leaving out the rate of return. People can very rationally choose a high risk for a high return. And the value of a return is always subjective, even when it is monetized (a single $1000 return means a lot to a broke person the day before rent is due, where a $2000 return five days later might have much lower value. Winning $5 off your worst enemy may be many times as satisfying as winning $500 off someone you have never met....)

It is clearly fair for Person A to disagree with Person B's subjective sense of probability or rate-of-return, especially if there are objective factors driving them that are clearly not being taken into account by Person B.

It is also possible that Person A can observe habits of thought in Person B that distort their sense of probability and subjective rates of return, and cause them to make decisions about risks that Person B themselves are likely to disagree with later.

But in an absolute sense, no, Person A cannot know that Person B is being irrational. He never has all the facts necessary, because payoffs are complex to the point of being paradoxical, and probabilities are influenced by very subtle details.

So is it fair for Person A to say Person B is making an irrational choice, if the subject is probabilities? – elliot svensson – 2018-04-09T18:59:22.293

"Being irrational" stands more like "act far from optimal", doesn't this sound more appropriate? Also "good for you" is dependent on person's values. And not only the probability of "good for you" but also the quantity is relevant. – rus9384 – 2018-04-09T20:33:46.680

Definitions of terms are off-topic on this SE. "Irrational" and "illogical" are only synonyms when taken in the colloquial sense, your issue may be in interpreting "illogical" in some more precise sense, like formal logic. Cost/benefit analyses, including probabilities, are part of informal logic and argumentation, broadly understood. – Conifold – 2018-04-09T20:59:31.947

IMO, the question is somewhat loaded. In my experience, people don't use the term "risk" objectively; choices are more likely to be called risky

becausethe speaker thinks its irrational. – None – 2018-04-09T21:15:44.650There are many ways to understand 'rational'. Do you mean 'is it prudent' or 'is it defensible within some formal system of logic'? – JeffUK – 2018-04-09T23:24:53.903