Boy or Girl Paradox - Bayesian Analysis

Bayesian Analysis

Following classical probability arguments, we consider a large Urn containing two children. We assume equal probability that either is a boy or a girl. The three discernible cases are thus: 1. both are girls (GG) - with probability P(GG) = 0.25, 2. both are boys (BB) - with probability of P(BB) = 0.25, and 3. one of each (G.B) - with probability of P(G.B) = 0.50. These are the prior probabilities.

Now we add the additional assumption that "at least one is a girl" = G. Using Bayes Theorem, we find

P(GG|G) = P(G|GG) * P(GG) / P(G) = 1 * 1/4 / 3/4 = 1/3.

where P(A|B) means "probability of A given B". P(G|GG) = probability of at least one girl given both are girls = 1. P(GG) = probability of both girls = 1/4 from the prior distribution. P(G) = probability of at least one being a girl, which includes cases GG and G.B = 1/4 + 1/2 = 3/4.

Note that, although the natural assumption seems to be a probability of 1/2, so the derived value of 1/3 seems low, the actual "normal" value for P(GG) is 1/4, so the 1/3 is actually a bit higher.

The paradox arises because the second assumption is somewhat artificial, and when describing the problem in an actual setting things get a bit sticky. Just how do we know that "at least" one is a girl? One description of the problem states that we look into a window, see only one child and it is a girl. Sounds like the same assumption...but...this one is equivalent to "sampling" the distribution (i.e. removing one child from the urn, ascertaining that it is a girl, then replacing). Let's call the statement "the sample is a girl" proposition "g". Now we have:

P(GG|g) = P(g|GG) * P(GG) / P(g) = 1 * 1/4 / 1/2 = 1/2.

The difference here is the P(g), which is just the probability of drawing a girl from all possible cases (i.e. without the "at least"), which is clearly 0.5.

The Bayesian analysis generalizes easily to the case in which we relax the 50/50 population assumption. If we have no information about the populations then we assume a "flat prior", i.e. P(BB) = P(GG) = P(G.B) = 1/3. In this case the "at least" assumption produces the result P(GG|G) = 1/2, and the sampling assumption produces P(GG|g) = 2/3, a result also derivable from the Rule of Succession.