Sanderson M. Smith

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The questions below appeared on the AP Statistics ListServe. They represents a very common misconception about probability, and, in this particular case, coin-flipping.

• Why doesn't the (absolute value of the) difference between the number of heads and the number of tails after a prolonged series of flips tend towards zero? I ran a simulation on Minitab and used successively larger sample sizes(500, 1000, 3000) and the differences were 15, 18 and 68? Why?

Response: It is true that if you flip a coin many, many times, the proportion of heads and the proportion of tails should both approach 50%. Note, however, that the writer is talking about differences, not proportions. I'll illustrate below with made-up numbers. (I didn't do a simulation.) Note that the difference between heads and tails continues to get larger, but the respective proportions approach 50%.

Somewhat related to the above: A few years ago, a parent of one of my students asked me this question.

• Suppose I flip a coin 100 times and get 40 heads and 60 tails. Is it reasonable to assume that if I flip the coin another 100 times, then I should expect to get more heads than tails because after 200 flips, I should get around 100 heads and 100 tails?

Note that the question-asker is thinking that the difference between number of heads and number of tails should approach zero after many flips. I know that the parent who asked the question is extremely intelligent. I did point out that the conclusion he reached would, in one sense, require that the coin had some kind of a memory. That is, it would have to realize that it would have to come up with more heads than tails during the next 100 flips in order to reach the expected number of heads (100) and the expected number of tails (100) after 200 flips.

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