Hello Mr. Gottlieb,
This is my first time writing in with a solution to Puzzle Corner, though
my fellow MIT alums and I often race to solve the problems with each new
issue, and it's a great way to keep in touch with those friends who love
puzzling.
I wanted to share a visual explanation of the Martin Gardner Tuesday
problem. It is a famous problem so I imagine many readers had already
encountered it, but most explanations that I have seen online have the math
correct but do little to explain *why *the extra information changes the
odds.
I think the grid below makes it much easier to understand. The problem
space can clearly be defined by a 2x2 grid, with the 4 possibilities: BB,
BG, GB, GG. For the first problem, when you learn that one child is a boy,
you can immediately rule out the GG case. This leaves three cases: BB, BG,
GB, and of those three, there is one case where both children are boys,
hence the probability of 1/3 that the other child is a boy. Most readers
are probably already familiar with this logic. The reason that the answer
is not 1/2 is because there is a case where *both* children meet the
specified condition, the BB case.
The same logic applies when you learn that one child is a boy, born on
Tuesday. Now we must carve up each cell of the 2x2 grid into a further 7x7
grid, representing the day of the week that each child was born on. This
creates a total of 196 possible combinations: 2x2x7x7 = 196.
[image: Inline image 1]
When you learn that one child is a boy, born on Tuesday, you can cross out
all the cases that do not meet these conditions. In the diagram below,
there is an "x" in every cell that does not include a boy born on Tuesday.
We are left with a total of 7+7+7+6 = 27 cells still possible. To answer
the question, we need to know how many of these possible cells have two
boys; these are represented by the dark gray shading in the top left box, a
total of 13 cells. Therefore, once you know that one child is a boy born on
Tuesday, the probability that the other child is also a boy is 13/27.
Just as in the previous case, the reason that the probability is not 1/2 is
that there is a duplicate case where *both* children meet the specified
condition, a boy born on Tuesday. While there is always only one duplicate
case, there are now many more non-duplicate cases, so the answer is now
much closer to 1/2. From here, it is easier to understand even more
specific cases. For example, if you know one child is a boy born on May 20,
the odds that the other child is a boy are now VERY close to 1/2.
Conceptually, this also makes some sense because the more specific
information you have about one child, the closer this situation becomes to
two distinct, independent events - because the likelihood that both
children meet the condition is so small, that case plays a smaller and
smaller role in the determining the probablity for the other child's gender.
I hope you find this to be a helpful and new way to think about a classic
problem. I really enjoyed thinking it through and coming to a fresh
approach.
Sam Ribnick
Class of '05, Course 8
--
Sam Ribnick
sribnick@gmail.com