Arguing the Toss.

And somewhat surprisingly, Cambridge have won the toss.

         Harry Carpenter

Sometimes you need a fair coin to toss so that you can make a choice between two options without any bias....Now suppose that the only coin you have available is a biased one: it does not have an equal probability (of 1/2) of falling 'heads' or 'tails'....Is there anything you can do in order to ensure that tossing a biased coin creates two equally likely, unbiased outcomes?
   Suppose that you toss the coin twice and ignore outcomes where both outcomes are the same - that is, toss again if the sequence 'heads-heads' (HH) or 'tails-tails' (TT) happens. There are two pairs of outcomes that could result: 'heads' followed by 'tails' (HT), or 'tails' followed by 'heads' (TH). If the probability of the biased coin coming down 'heads' is p, then the probability of getting 'tails' is 1 - p, and so the probability of getting the sequence HT is p(1 - p) and that of getting TH is (1 - p)p. These two probabilities are the same, regardless of the probability p of the biased coins. All we have to do to get a fair game is define HEADS by the sequence HT and TAILS by the sequence TH, and the probability of TAILS is the same as the probability of HEADS. And you don't need to know the bias, p, of the coin.*


* This trick was thought up by the great mathematician, physicist and computer pioneer, John Von Neumann. It had wide use in the construction of computer algorithms. One of the questions that was subsequently addressed was whether there were more efficient ways of defining the new HEAD and TAIL states. The way we have done it wastes 'time' by having to discard all the HH and TT outcomes.

John D. Barrow



If you're worried by the expression 'p(1 - p)', remember that in Probability Theory probabilities are expressed as decimal fractions between 0 and 1, so that 0 represents impossibility, and 1, certainty. Thus, if a coin is rigged to fall 'heads' 6 out of every 10 times tossed, the probability, p, of H is 0.6 (and, clearly, p of T is 0.4)
    In our example,

           p(1 - p)

     or,  p - p^2 (representing p squared)

      =   0.6 - 0.6^2

      =   0.6 - 0.36

      =   0.24


Von Neumann was a character. His work on Game Theory, and his membership of the United States Atomic Energy Commission, mixed with a hatred of the Soviet Union, led him to develop the wonder that was the equilibrium strategy he called, with deliberate humor, mutually assured destruction, or MAD. He described himself, before Senate Committee, as "violently anti-communist, and much more militaristic than the norm." And, elsewhere, "If you say why not bomb [the Soviets] tomorrow, I say, why not today. If you say today at five o'clock, I say why not one o'clock."