The importance of a flag
Metalist Stadium, Kharkiv, June, 13th 2012, 21:45 p.m.. It's going to take place in the city of Kharkiv (Ukraine) the European Championship football match between the teams from Germany and Netherlands.
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vs. | 
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The stadium is completely full of fans from both countries.
The number of German supporters is equal to the Dutch ones, and moreover,
they are uniformly distributed and mixed in the stands, as the organization considers
that it's a low-risk match.
As soon as it ends the first part of the match, the fans take the opportunity to go to the toilet. The bathrooms are are located by pairs (ladies' and gentlemen's) all around the stadium. We approached one pair of them, and we note that they are occupied.
Could you tell me which is the probability that both toilets are
occupied by German fans?
And if, next to the ladies' room, we see a German flag belonging
to its occupant, but we don't know who stands at the men's room,
which would be now the probability?
Finally, if the German flag is at the same distance from both doors,
so that we can't know who its occupant is, which would be now the probability
that both occupants were German?
Go to the second half to discover the solution
SECOND HALF
To solve this problem, we'll use the following table, in which we can see the four different possible 'ocuppations' of both toilets, men's and ladies':
| Gentlemen's | Ladies' | |
|---|---|---|
| Option 1 | German | German |
| Option 2 | Dutch | German |
| Option 3 | German | Dutch |
| Option 4 | Dutch | Dutch |
In the first case, there's no flag to help us identify the occupants, so we have 4 possible ways to occupate the bathrooms. But only in one option (table's option 1), both toilets are occupied by German fans. Therefore, the probability is 1 / 4 = 25%.
In the second case, logic tells us that if the ladies' toilet is occupied by a German supporter, and if in the men's one we can find either a German or a Dutch supporter, it's clear that probability will be of 50%. And of course, we have 2 possible options (table's options 1 and 2) that comply with the requirement that the ladies' is occupied by a German supporter, and only one of them (table's option 1), in which both occupants are German. So, the probability in this case amounts to 1 / 2 = 50%.
The curious thing happens when we place the German flag between the two doors. This indicates that there's, at least, one German fan occupying one of the toilets. Here, we are tempted to conclude that the probability should be the same as in the previous case, but, contrary to what apparently seems like common sense, the solution is very different.
If we look at the table, we can see that in this case we find three possible options in which at least one toilet is occupied by Germans (options 1, 2 and 3), and there's only one favorable case (case 1). So the probability would be 1 / 3 = 33.3%.
Surprisingly, and contrary to what our intuition tells us to, in this case the probability decreases to one third.
This subject shows us that, for any problem, in spite of its apparently simplicity, it's always better to take care accurately of what the possible cases are and the favorable ones involved on it, just to get right to the solution.
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Return to the first half
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