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.

They're Still Getting It Wrong

It is a well known fact that boys have a higher variance on high school math tests than girls. No one knows what causes this - maybe there are different social pressures on boys, maybe boys test differently, maybe boys are innately different. Anyways, the results of the latest study which, again, showed higher male variance are still being misreported:

26. Girls and boys now perform equally on standardized high school mathematics tests across North America, ending a gender gap that lasted for decades.

The source for this quote is the study I linked to, which showed that the extreme scores are more likely to be male's scores. The study showed major differences in the scores! Of the major media outlets, only the WSJ got it right the first time around. Of course the bad information was passed on.



What shocks me about all of the coverage is that no one has reported the differences between whites' scores and Asian-Americans' scores in the study. While the results from white Americans fell in line with previous results, namely that twice as many boys as girls were expected in the 99th percentile on math tests, this wasn't true for Asian-Americans. An Asian-American female was actually a hair more likely to be in the top percentile than an Asian-American male, though this fact didn't hold for the top five percent of scores. I haven't seen quantitative studies on differences in values across different racial groups, but from anecdotes and personal experience I think that Asian-Americans parents have fewer differences in expectations from daughters and sons. This is the first data (I've seen) which supports the variance difference being an environmental one. There wasn't enough data on any other group to make it into the study.