Last week the European Decision professionals (
EDPN) had its first conference. Theme of the conference was competing through quality of strategic decisions.
Reidar Bratvold, one of the speakers, touched upon the fact that we as humans are unfit to deal with uncertainty in reasoning. The examples he used, reminded me of a book that I recently read, The Drunkard's Walk: How Randomness Rules Our Lives by Mlodinow. Bratvold had asked decision professionals in Oil and Gas industry about a well-known puzzle in statistics which is also in Mlodinow’s book. It goes like this:
You are told that a family, completely unknown to you, has two children and that one of the children is a daughter. What is the chance that the other child is also a daughter?
About 80% of the people Bratvold asked responded that the chance is equal to ½. Bratvold repeats the question making explicit that no statement whatsoever is made on the birth order of the children. With that remark, still 80% of the people responded that the chance is equal to ½.
The correct answer however is 1/3. With two children in a family there are obviously four possibilities: {Girl, Girl}, {Girl, Boy}, {Boy, Girl} and {Boy, Boy}. Since one of the children is a girl, the {Boy, Boy} possibility must be eliminated. That leaves us with 3 possible options. The chance that the other child is a girl as well therefore is 1/3. To come to the correct answer I used
Cardano’s method. As he explains in “Book on Games of Chance” it is best to construct a sample space to calculate the odds of even the simplest events. Trust your instincts instead and you’re bound to go wrong, even the decision professional.
To test your instincts, what if we add a seemingly irrelevant remark to the above question? Suppose that the daughter is born on a Friday. What are the chances of both children being girls?
Let’s follow Cardona’s advice and built a sample space. We had three possible combinations, {Girl, Girl}, {Girl, Boy}, and {Boy, Girl}. For each combination count the possible combinations of weekdays of birth:
{Girl, Girl} = {Mon, Fri}, {Tue, Fri}, {Wed, Fri}, {Thu, Fri}, {Sat, Fri}, {Sun, Fri}, {Fri, Mon}, {Fri, Tue}, {Fri, Wed}, {Fri, Thu}, {Fri, Sat} {Fri, Sun}
- {Boy, Girl} = {Mon, Fri}, {Tue, Fri}, {Wed, Fri}, {Thu, Fri}, {Fri, Fri}, {Sat, Fri}, {Sun, Fri}
- {Girl, Boy} = {Fri, Mon}, {Fri, Tue}, {Fri, Wed}, {Fri, Thu}, {Fri, Fri}, {Fri, Sat}, {Fri, Sun}
The {Fri, Fri} outcome for the combination of two girls is not valid, because we know that only one girl was born on a Friday. In total there are 26 possible outcomes of which 12 are favorable. So the chance of two girls in this case becomes 6/13.
My and probably your intuition as well would have been to discard the additional information on the weekday of birth as irrelevant. But it isn’t! The probability even rises because of it, with whopping 38%! Imagine the impact of this is when it comes to real world decisions like exploring new oil fields, medical decisions or investing in the development of new drugs. So my advice would be before you entrust your daughter to a decision professional to ask him if is knows Gerolamo Cardano. As irrelevant that question may be, it can have a large impact.
