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Coin Flipping is Biased

7 March 2004

Persi Diaconis, Mary V. Sunseri Professor of Statistics and Professor of Mathematics at Stanford University, delivered a talk at the 170th Annual Meeting of the American Association for the Advancement of Science (AAAS) in Seattle, February 12-16, entitled "The Search for Randomness". According to the AMS report on the meeting, his talk was

"...organized around Ten Interesting Things About Flipping a Coin. Perhaps the most interesting among the ten is that Diaconis, Richard Montgomery, and Susan Holmes found that a coin toss is biased towards the side that faced up when the coin was tossed. In fact, they found that the probability that a coin lands the same way that it was facing before the flip is over .51 . Coin-tossing is a basic experiment, which is assumed to be random. It is surprising that this assumption could be false and Diaconis concluded his talk by cautioning the audience to look closely at assumptions when probability and statistics are applied to real-world problems. In response to a question, Diaconis demonstrated that after years of practice he could control the outcome of a coin flip, but being a good magician, he would not reveal how."

You can read more about the work by Diaconis, Holmes and Montgomery in Science News On Line Week of Feb. 28, 2004; Vol. 165, No. 9 , p. 131. Further news on Persi Diaconis and his personality and interesting background may be found here. There he is reported as saying:

"I work from seven A.M. to midnight each day. I'm always doing mathematics."

That line reminded of the following paragraphs I once read in Hamming's "You and Your Research":

Now for the matter of drive. You observe that most great scientists have tremendous drive. I worked for ten years with John Tukey at Bell Labs. He had tremendous drive. One day about three or four years after I joined, I discovered that John Tukey was slightly younger than I was. John was a genius and I clearly was not. Well I went storming into Bode's office and said, ``How can anybody my age know as much as John Tukey does?'' He leaned back in his chair, put his hands behind his head, grinned slightly, and said, ``You would be surprised Hamming, how much you would know if you worked as hard as he did that many years.'' I simply slunk out of the office!

What Bode was saying was this: ``Knowledge and productivity are like compound interest.'' Given two people of approximately the same ability and one person who works ten percent more than the other, the latter will more than twice outproduce the former. The more you know, the more you learn; the more you learn, the more you can do; the more you can do, the more the opportunity - it is very much like compound interest. I don't want to give you a rate, but it is a very high rate. Given two people with exactly the same ability, the one person who manages day in and day out to get in one more hour of thinking will be tremendously more productive over a lifetime. I took Bode's remark to heart; I spent a good deal more of my time for some years trying to work a bit harder and I found, in fact, I could get more work done. I don't like to say it in front of my wife, but I did sort of neglect her sometimes; I needed to study. You have to neglect things if you intend to get what you want done. There's no question about this.

Whilst I agree with the spirit of what Hamming says, I am not afraid to say that working seventeen hours a day sounds like a lot to me. The only way to do so would be to devote your entire life to your work, and not to have a family and/or other hobbies. This may be worth it for somebody exceptional like Persi Diaconis, but I feel not for an average guy like me.