## Flawed analysis of “one child is a boy” problem?

A mathematical puzzle has reappeared over the last year as the topic of discussion in various blogs and I have not seen any discussion suggesting that the analysis appearing in blogs contains a fundamental flaw.

The problem is as follows: I have two children and at least one of them is a boy; what is the probability that I have two boys? (A variant of this problem specifies whether the boy was born first or last and has a noncontroversial answer).

Most peoples (me included) off-the-top-of-the-head answer is 1/2 (the other child can be a girl or a boy, assuming equal birth probabilities {which is very a good approximation to reality}).

The analysis that I believe to be incorrect goes something like the following: The possible birth orders are `gb`

, `bg`

, `bb`

or `gg`

and based on the information given we can rule out girl/girl, leaving the probability of `bb`

as 1/3. Surprise!

A variant of this puzzle asks for the probability of boy/boy given that we know one of the children is a boy born on a Tuesday. Here the answer is claimed to be 13/27 (brief analysis or using more complicated maths). Even greater surprise!

I think the above analysis is incorrect, which seems to put me in a minority (ok, Wikipedia says the answer could sometimes be 1/2). Perhaps a reader of this blog can tell me where I am going wrong in the following analysis.

Lets call the known boy `B`

, the possible boy `b`

and the possible girls `g`

. The sequence of birth events that can occur are:

`Bg gB bB Bb gg`

There are four sequences that contain at least one boy, two of which contain two boys. So the probability of two boys is 1/2. No surprise.

All of the blog based analysis I have seen treat the ordering of a known boy+girl birth sequence as being significant but do not to treat the ordering of a known boy+boy sequence as significant. This leads them to calculate the incorrect probability of 1/3.

The same analysis can be applied to the “boy born on a Tuesday” problem to get the probability 14/28 (i.e., 1/2).

Those of you who like to code up a problem might like to consider the use of a language like Prolog which I would have thought would be less susceptible to hidden assumptions than a Python solution.

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