(for 3 Quarks Daily)
“You are aware”, I ask a pair of students celebrating their fourth successful die roll in a row, “that you are ruining this experiment?” They laugh obligingly. In four pairs, a small group of students is spending a few minutes rolling dice, awarding themselves 12 euros for every 5 or 6 and ‘losing’ 3 euros for every other outcome. I’m trying to set them up for the concept of expected value, first reminding them how to calculate their average winnings over several rounds, and then moving on to show how we calculate the expected average without recourse to experiment. It would be nice, of course, for their experimental average to be recognizably close to this number. Not least since this particular lesson is being observed by the Berlin board of education, and the outcome will determine whether or not I can get a teaching permit as a foreigner.
In case they are reading this, I would like to emphasize that I plan all my lessons with care and forethought; but for this particular one, you can bet I prepared especially well and left nothing to chance. Except for the part I left to chance, that is. To be precise: I had neglected to calculate in advance how likely it was for the experimental average over roughly 80 games to diverge from the expected value by a potentially confusing amount. I relied on my intuition, which informed me that 80 is a large number.
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