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My previous post described how this puzzle doesn’t give enough information to be definitively solved. We need to know the mother’s strategy for revealing the gender of the first child.

In that earlier post, I approached things from the perspective of repeated simulations, giving some code to demonstrate. Here’s a more visual argument.

My thinking is that in order to make a judgment about the likeliness of different outcomes, we need to know the probability tree for the different events: the possible genders of the children and which child is revealed first. We can deduce almost the entire tree:

But we are never told how the mother decides which child to reveal when she has both a boy and a girl.

In my last post, I gave two ways of completing this tree:

• If the mother always reveals the gender of the female first whenever possible:

  

In this case, the a priori chance of the second child being a boy is higher.

• If the mother always randomly decides which child she will reveal the gender of first:

  

  In this case, the a priori chance of the second child being a boy is 50%.

So, to answer the question using a given a probability tree, we examine the possible states we can be in (states in which the first child revealed is ‘G’). Then we prune off the other states and refigure the odds using the rules of conditional probability.

So, depending on which tree is being used, the answer to the question is different.

• A girl is always chosen first if possible

  

  Probability of a boy second = 2/3

• First child is always chosen randomly

  

  Probability of a boy second = 1/2

This entry was posted on Friday, January 2nd, 2009 at 4:20 pm and is filed under math. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.