$$P(X&Y) <A \and B) \leq P(X)
The conjunction fallacy is consists in assuming that specific conditions are more probable than more general ones.
For the wayreasons related to representativeness heuristic, a fleshed-out story can soundthat contains typical amount of detail sounds more plausible if you add more detailsthan a stripped-down description of a situation that only states a few facts. There is a tendency for people to it, even though in reality every extra detail makes the whole story strictly less probable.
For example:
The statement "Bill is an accountant who plays jazz for a hobby." is less probable than the statement "Bill plays jazz for a hobby."
Intuitively, you might expecttake that Bill is more likely to be an account who plays jazz, than he is to play jazz.
plausibility at face value, and assign probability accordingly. This intuition is wrong, because the conjunction rule of probability theory states that, for all X and Y, any event X, its conjunction with additional details Y will be less probable:
P(X&Y) <= P(Y).X)
The conjunction fallacy suggests that one should be very careful in adding details to any claim, as even though each such detail may make the claim so much more convincing, it also inevitably subtracts from its validity.
Every detail thatThe conjunction fallacy is the way a story can sound more plausible if you add more details to a story will always makeit, even though in reality every extra detail makes the whole story strictly less probable, even if it makes the story sound more plausible.probable.
Every detail that you add to a story will always make the whole story less probable, even if it makes the story sound more plausible.
For example:
The statement "Bill is an accountant who plays jazz for a hobby." is less probable than the statement "Bill plays jazz for a hobby."
Intuitively, you might expect that Bill is more likely to be an account who plays jazz, than he is to play jazz.
This intuition is wrong, because the conjunction rule of probability theory states that, for all X and Y, P(X&Y) <= P(Y).
For the reasons related to representativeness heuristic, a fleshed-out story that contains typical amount of detail sounds more plausible than a stripped-down description of a situation that only states a few facts. There is a tendency for people to take that plausibility at face value, and assign probability accordingly. This intuition is wrong, because the conjunction rule of probability theory states that, for any event X, its conjunction with additional details Y will be less
probable:$$P(A \and B) \leq P(B)$$
byEliezer YudkowskybyEliezer Yudkowsky