They learned something useful!

If you toss a coin twice, half the time you will get either all heads or all tails. If you toss it four times, only an eighth of the time will there be all heads or all tails. When you toss a coin ten times, it is all all the same only once in 512 times. 98% of the time there are at least two heads and two tails with 10 tosses.

If something is done repeatedly and has the same probability of happening each time, the more times you do it, the more likely it is to be close to the average value. This is called the law of large numbers. To carry the coin problem a bit further, if a coin is tossed 1000 times, there is a 99.8% chance there will be between 450 and 550 heads.

The lottery and Los Vegas work on this principle. All the games are designed to give the house an advantage. Sometimes the advantage is as low as 5.3%, but in the long run the house wins. The law of large numbers guarantees it. To quote Gilbert in a song about roulette:

Where’er at last the ball pops in, The bank is bound to win!

I gave my probability class a questionnaire about my teaching, which I read after posting my grades. Two of my students answered the question, “What did you learn best in this course?” with the statement that they shouldn’t gamble. I am pleased I taught them something useful.

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