Conjunction fallacy I was reading this article which has the following question, > Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. > > Which is more probable? > > Linda is a bank teller. > Linda is a bank teller and is active in the feminist movement. I chose the second option but according to a set theorem the choice was wrong. Why is not possible that linda can be both a bank teller and feminist? What is the simple explanation to this with example?

Let $A$ be the event "she is active in the feminist movement" and let $B$ be the event "she is a bank teller." Then "she is active in the feminist movement **and** is a bank teller" is $A\cap B$. Whatever opinions one might have about whether bank tellers might or might not have a tendency to be active in the feminist movement, it is for certain true that $A\cap B \subseteq B$. Thus it is _automatic_ that $$P(A\cap B) \le P(B).$$ So we **cannot** have $P(A\cap B) \gt P(B)$. It is _logically impossible_. Information that we are provided about Linda's background and history cannot alter that fact.