Intended for healthcare professionals

Endgames Statistical question

Conditional probabilities

BMJ 2009; 338 doi: https://doi.org/10.1136/bmj.b113 (Published 15 January 2009) Cite this as: BMJ 2009;338:b113

The advantage of expressing probabilities as natural frequencies

Is John Fletcher's argument to be called fallacious or just poorly
formulated? It is somewhat similar to the "Monty Hall problem" analysed
e.g. by Gerd Gigerenzer [1]. Let us apply Gigerenzer's recommendation for
clear thinking: formulate problems in terms of absolute frequencies rather
than probabilities for an individual case.

Fletcher could have formulated
his problem this way: "Consider 1000 families having exactly two children,
with both of them wearing gender-ambiguous names such as Pat and Alex. Now
among those families where one child is identified as female, in how many
is the other child expected to be boy?" This makes his argument much
clearer: about 750 families can be expected to have at least one girl and
among them, in about 500 is the other child expected to be a boy; hence
the right answer is 500/750 or 67% [d]. Conversely, Pelham Barton's
argument may be stated as follows: "Consider 1000 two-children families
where one child is identified as a female, in how many is the other child
expected to be male?" Here, the presence of one girl among the siblings is
no more a "filter" to select a subset out of the initial population, it
rather belongs to the initial population definition itself. As the gender
of the second child is independent from the first one, the correct answer
is now 500/1000 or 50% [a]. Expressing questions in terms of absolute
frequencies forces us to make explicit choices about the numerator and to
disclose selection biases introduced by the provision of additional
knowledge, even apparently unrelated to the outcome.

1. G. Gigerenzer, Reckoning with risk, Penguin Books, London 2003, pp
217-223.

Competing interests:
None declared

Competing interests: No competing interests

21 January 2009
Thierry Buclin
MD, clinical pharmacologist
University Hospital CHUV, 1011 Lausanne, Switzerland