Randomizing a Coin Toss

Way to report only the problem, Freakonomics blog:

But it may be that the the random coin toss isn’t so random. A 2007 study found that a vigorously flipped coin is likely to land on the same side it started on at least 51 percent of the time, possibly more depending on the person doing the flipping.

Say you have a coin with a bias that doesn't change from flip to flip. You can remove the bias by flipping the coin twice, and taking the first result if the second result differs from the first or repeating if it lands the same way both times.

Mathematically, if the probability of landing heads is P, then:

• P^2 = Prob[HH]
  
• P*(1-P) = Prob[HT]
  
• (1-P)*P = Prob[TH]
  
• (1-P)^2 = Prob[TT]
  

Since the probability of HT is the same as TH, taking the first element removes the bias.

Mad props to von Neumann for inventing this trick (publication was "Various techniques used in connection with random digits"), along with fundamental portions of the theory for quantum mechanics, cryptography, and game theory. They don't make mathematicians like they used to.