The Tuesday Boy Problem

The Tuesday Boy Problem


I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?

My first reaction after reading this in the ScienceNews website was: “what does tuesday have to do with the problem?”, or in other words: “In what way could the information on the day of birth affect the probability?”.

At first glance, the information on the day of birth seems irrelevant indeed- how could the birth day of the first child possibly affect the sex of the second child? Since the problem statement does not clearly specify the sex of the younger kid, one of the following three events could happen:

Boy, Girl

*

Girl, Boy

*

Boy, Boy

Hence, the probability to have two boys seems to be 1/3.

The solution might seem obvious, and one might even rush into concluding that a solution to this problem is trivial.

But let’s try to approach the question from a different angle. Suppose you were involved in a survey, and your task was to reach by phone families that have two children, with one child a boy born Tuesday. Would the extra constraint on the birth date limit the number of families which are relevant? Clearly, the “Tuesday”  information will reduce the number of families relevant to the survey. To make this even more clear, one could think of adding an extra constraint on the month of birth, or on the year of birth, etc … There is clearly going to be an impact on the probability and thus on the answer.

Accounting for the day of birth, one can enumerate the possible cases more carefully in the following way:

Boy born on a Tuesday, Girl born on a Monday

Boy born on a Tuesday, Girl born on a Tuesday

Boy born on a Tuesday, Girl born on a Wednesday

Boy born on a Tuesday, Girl born on a Thursday

Boy born on a Tuesday, Girl born on a Friday

Boy born on a Tuesday, Girl born on a Saturday

Boy born on a Tuesday, Girl born on a Sunday

*

Girl born on a Monday, Boy born on a Tuesday

Girl born on a Tuesday, Boy born on a Tuesday

Girl born on a Wednesday, Boy born on a Tuesday

Girl born on a Thursday, Boy born on a Tuesday

Girl born on a Friday, Boy born on a Tuesday

Girl born on a Saturday, Boy born on a Tuesday

Girl born on a Sunday, Boy born on a Tuesday

*

Boy born on a Tuesday, Boy born on a Monday

Boy born on a Tuesday, Boy born on a Tuesday

Boy born on a Tuesday, Boy born on a Wednesday

Boy born on a Tuesday, Boy born on a Thursday

Boy born on a Tuesday, Boy born on a Friday

Boy born on a Tuesday, Boy born on a Saturday

Boy born on a Tuesday, Boy born on a Sunday

*

Boy born on a Monday, Boy born on a Tuesday

Boy born on a Wednesday, Boy born on a Tuesday

Boy born on a Thursday, Boy born on a Tuesday

Boy born on a Friday, Boy born on a Tuesday

Boy born on a Saturday, Boy born on a Tuesday

Boy born on a Sunday, Boy born on a Tuesday

The probability can then be simply computed as the number of  cases where the younger and older kids are both boys ( = 13) by the total number of possible cases ( = 27 ), i.e. 13/27.

I chose to write on this problem because it challenged my intuition, which, used alone, did not lead to the right answer. Some argue that we should more often listen to our intuition as it can hold tremendous power in decision making and problem solving. This can be true in some cases, but it is clearly not true for the “Tuesday boy” problem.

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