The `Conjunction Fallacy’ is a fallacy or error in decision making where people judge that a conjunction of two possible events is more likely than one or both of the conjuncts. A good description can be found here.

The most famous example is due to Tversky and Kahneman (1983), where they gave the following scenario:

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?

a) Linda is a bank teller.
b) Linda is a bank teller and is active in the feminist movement.

The correct answer, according to the laws classical probability theory, is a). Nevertheless, answer b) seems intuitively more reasonable, and indeed experiments have demonstrated that many people choose b) over a).

To understand this behaviour we need to construct a model of reasoning that isn’t based on classical probability. A model based on quantum theory was given by Busemeyer et al. (2011). In this model the events “Feminist” and “Bank Teller” are represented by a pair of non-orthogonal bases, and the initial knowledge state of a decision maker, after having read the vignette, by a state vector in this space. Depending on the relative angles between the basis vectors and the state vector we can reproduce the experimentally observed probabilities.

The example below lets you adjust the angle between Bank Teller and Feminist, and the angle of the state vector, and computes the various probabilities. It can also give a visual indication of how the projections from the initial state onto the various possibilities work.

[WolframCDF source=”https://cdn.vanderbilt.edu/vu-my/wp-content/uploads/sites/2080/2016/04/14134423/Con_Fal22.cdf” width=”500″ height=”615″ altimage=”” altimagewidth=”” altimageheight=””]