The conjunction fallacy consists in assuming that specific conditions are more probable than more general ones.
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.
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.