# Probability Riddle

mfb
Mentor
I drew two pens, both times I had a 50% probability to get a black pen and a 50% probability to get a blue pen.
At least one of the pens was black. What is the probability the other one was blue?

You can use the standard answer of 2/3 - and be wrong.
Why? Before I drew the pens, I decided to post this puzzle only if the colors are matching. There was never a chance of mismatching colors for this puzzle.

The frequentist approach does not work here as the children are either boys or girls (each) - there is no actual probability involved. It is possible to take the Bayesian approach - but then we also have to ask "given the different girl/boy combinations, how likely is it to get the information 'at least one is a boy born on Tuesday'?". And there is no meaningful way to assign a specific probability to that.

PeroK
Homework Helper
Gold Member
The frequentist approach does not work here as the children are either boys or girls (each) - there is no actual probability involved. It is possible to take the Bayesian approach - but then we also have to ask "given the different girl/boy combinations, how likely is it to get the information 'at least one is a boy born on Tuesday'?". And there is no meaningful way to assign a specific probability to that.
You are correct, of course. If someone volunteers some information, you've really no way of knowing why they have told you that precisely.

The moral is that the information for these problems has to be obtained by direct questions and not volunteered.

That's why I stated a more precise interpretation of the original problem above: pick a child at random and state the day of the week on which it was born.

It is tempting to solve this riddle by constructing a 14x14 grid, finding 27 cells where there is a Tuesday boy, and 13 of them have two boys.

But you need a 14x14x2 grid. The third dimension is for what the parent tells you. A parent with a Tuesday boy could tell about a Thursday girl. Using this grid, 28 cells tell you about a Tuesday boy, and 14 of them have two boys.

For historians, this is a variation of Bertrand Box Paradox. The paradox is that if the probability is X for every combination of gender a d day, it must be X regardless of gender and day. And this X must be 1/2.

Better one:
What is the probability of getting this question right?
A:25%
B:50%
C:60%
D:75%

Better one:
What is the probability of getting this question right?
A:25%
B:50%
C:60%
D:75%
E: Less than 1% (see below), because we confuse English meaning with Math meaning.

In English, the word "given" means information that is given to you, and so must apply to every remaining possibility.That is, the information is a necessary condition: If an outcome is still possible, then it fulfills this information.

In the mathematics of probability, the word "given" means more. It means an event (a set of possible outcomes) that describes every remaining possibility, and excludes every remaining impossibility. That is, the information is a both a necessary condition and a sufficient condition: If an outcome is still possible, then it fulfills this information; and if an outcome fulfills this information, then it is still possible.

Most paradoxes of this type arise from not considering how being a sufficient condition reduces the size of the event used as a condition. The most famous example is the Monty Hall Problem: on a game show, you choose one of three doors that conceal exactly one desirable prize. But the host doesn't open your door just yet. Instead, he opens one of the other doors that doesn't have the prize (he knows where it is, so he can always do this). He then offers to let you switch to the third door. Should you?

At first glance, most people say "it doesn't matter." They reason that the set of remaining possibilities includes 2/3 of the possible situations: the 1/3 chance where the prize was placed behind your door, and the 1/3 chance where the prize was placed behind the third door. This is wrong, because "the prize isn't behind Door X" is just a necessary condition for him to open Door X. If you originally chose the prize door, the host can open either of the others.

The correct solution - which is almost never given, and why I said "less than 1%" above - is that the set of remaining possibilities includes only 1/2 of the possible situations: all of the ones where the prize was placed behind the third door, but only half of those where the prize was placed behind your door. So your door has a (1/6)/(1/6+1/3)=1/3 chance of winning, and the third door has a (1/3)/(1/6+1/3)=2/3 chance.

The same reasoning applies to either the "Two Child Problem" (or the "Tuesday Boy Problem"). The answer 1/3 (or 13/27) is correct only if you have a reason to believe that every parent of a boy and a girl (or a parent of two with exactly on boy born on Tuesday) would tell you about the boy (or the boy born on Tuesday), If they might tell you something else, the answer is 1/2 in both cases.