Gary Foshee, a collector and designer of puzzles from Issaquah near Seattle walked to the lectern to present his talk. It consisted of the following three sentences: “I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?”
“The first thing you think is ‘What has Tuesday got to do with it?'” said Foshee, deadpan. “Well, it has everything to do with it.” And then he stepped down from the stage.
This is the answer: 13/27.
Many people will intuitively say that the answer is 1/2 (=the chance of having a boy or a girl), but probability aficionados will give the answer 1/3, since this is the Boy or Girl Paradox: We are not told that the speaker has a child and is waiting for another, but that he already has two children. Two children can come in four configurations: 1) boy/girl, 2) girl/boy, 3) girl/girl, 4) boy/boy. Since he has one boy, we are looking at the options 1, 2, or 4. Only the boy/boy combination includes two boys, so the probability is 1/3. In other words, order matters and completely changes probability.
So what has being born on a Tuesday got to do with it? Why would the answer not still be 1/3? The New Scientist has a good explanation toward the bottom of the article. Simply count the different combinations of genders and weekdays, which gives the result (number of combinations with two boys, at least one of which was born on a Tuesday) / (number of combinations with at least one boy born on a Tuesday). The result really is 13/27.
This is the best illustration I have found: This shows all the boy/girl pairs as well as the possible weekdays on which they could be born. Green represents situations with two boys, at least one of which was born on a Tuesday. Yellow represents at least one boy born on a Tuesday. Red is neither. Hence the answer is green/(green+yellow)= 13/(13+14) = 13/27.
But again, what has Tuesday got to do with it?
It is an appealing puzzle because even once you (or at least I) accept the 13/27 calculation and visualize it, it still is wildly counter-intuitive that being born on a Tuesday influences the probability of having a brother. (Actually it doesn’t – see below.) What’s also strange is that with a rarer trait (say, being born on February 29), the probability of having two boys approaches 1/2. (For a trait with probability 1/A, the probability of having two boys is 2A-1/4A-1.)
Other people with more skill in probability theory have explained it well (search for “cocktail party” here), but I thought about it moving backwards from the result: Our intuition (which is correct) is that boys born on Tuesdays do not have a larger chance of having a brother than boys born on other days. Therefore the 13/27 result seems absurd because we could repeat it with all days of the week, add up the results, and reach the conclusion that the answer to the boy or girl paradox was 13/27 and not 1/3. And that is the hint: We are not really calculating what we think we are calculating.
It is really a question of formulation. The puzzle is not “My firstborn is a boy born on Tuesday” or “I have exactly one boy born on a Tuesday”, but “One [or more] is a boy born on a Tuesday [but I am not telling you which one].” Which means that we did not calculate whether being born on a Tuesday influences the probability of having a brother.
As I stated above, the absurdity comes from the fact that we might imagine that we had sliced the possibility space (2 genders, 7 weekdays) into 7 equally big slices, but did we? Try repeating the calculation for “boy born on a Tuesday”, “boy born on a Wednesday” etc…
Now consider the case of the oldest child being a boy born on a Tuesday, and the youngest child being a boy born on a Wednesday. This will count toward both the probability of “two boys when at least one child is a boy born on a Tuesday” and “two boys when at least one child is a boy born on a Wednesday”. Many cases are counted for several weekdays. This is why the probability goes up from the expected 1/3, approaching 1/2.
Why this is such a good puzzle:
- It is a puzzle within a puzzle.
- Like the Boy or Girl Paradox, which encourages us to answer from the knowledge that the boy/girl probability is (roughly) 50%, this puzzle encourages us to answer from the knowledge that being born on a given weekday generally doesn’t influence other probabilities. In both cases, that is not what the puzzle is actually asking us. We are tricked into using a stock answer that is not appropriate for the given puzzle.
- It appeals to puzzle and probability theory fans by building on an existing puzzle.
- It gives a result (approaching 1/2) which is exactly the result that connoisseurs know should not be the answer.
- It uses a counting technique specifically designed to prevent from counting the same case twice, and we may be thinking that “at least I am not counting anything twice”, but that is exactly what we are doing.
- The very terse explanation of the puzzle is deliberately misleading. When Foshee says ‘One is a boy born on a Tuesday. … What has Tuesday got to do with it? Well, it has everything to do with it.“, it makes it sound like we are doing a probability calculation depending on one particular boy, when in fact we are doing a calculation depending on “one or more children being a boy born on a Tuesday” – hence we end up counting some combinations for several weekdays.