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Imagine spending the summer polishing your PhD thesis in the library. Every day you take the subway, a train going up North to the university.
One day you realize that the trains going South can bring you to the beach. Nothing is wrong with some leisure. You calculate that if you spend half of the remaining summer vacation in the library that should be enough to finish the thesis. To spice up the summer,every day you catch the first train that comes to the platform. It may be South train going to the beach or the North one heading to the university.
Since you come to the platform randomly in-between 9 and 10am, and the trains go on a regular schedule with the same frequency in both directions, you guesstimate you will end up spending about the same amount of time in the library and at the beach.
On the first week of this experiment you are surprised that chance brought you to the library only once a week. “This will correct itself during the summer, like a heads-and-tails game,” you think. But even after two months you find yourself at the library only 1/5th of the time– and the thesis at grave risk.
How could this happen?
(The answer is below. For your brain’s sake, try to solve it before checking out the answer!)
Maria Lando (aka TheMathMom) illuminates and demystifies mathematics for adults, describing its fun use in everyday life. TheMathMom has been featured in multiple media outlets including The Boston Globe, The Jewish Advocate, and Redbook Magazine. Learn more at TheMathMom.com.
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SOLUTION
Sadly, you ignored one crucial variable.
Let’s assume trains leave every 10 mins, with train to the beach departing from your station at 10 minutes past the hour, 20 mins, 30 mins etc while train to the library leaves exactly 2 mins after the beach train, at 12, 22, 32,.. minutes past the hour. Then, only if you come in-between 10 and 12, 20 and 22, 30 and 32 mins past the hour you you will wait and pick the library train. If you come in-between 12 and 20, 22 and 30, 32 and 40 mins past the hour you will wait and pick the beach train.
2 minute windows for the library train and 8 minute windows for the beach train.
As you arrive to the platform randomly in-between 9 and 10am, your chances of seeing the library train first are 2/10=1/5 and beach train first are 8/10=4/5. So, you end up finding yourself at the library on average only once out of 5‑day work week and at the beach 4 times during the 5‑day work week.
Random doesn’t always lead to half-and-half 