Numbers can Cause Confusion: Binomial Coefficients Can Help
Bad Science writer Ben Goldacre wrote about how “Guns don’t kill people, puppies do,” where he points out some ways in which the probabilities of things can be a bit confusing. An extremely common example, which was messed up by a British newspaper, is the probability of multiple people (siblings in this case) sharing a birthday.
Like he says, the trick is that it doesn’t matter when the first child was born, only that the next two were born on the same day. This is a trivial case in which the binomial coefficient simply cancels out the leading 1/365. In general, though, you can use the binomial coefficient (sometimes called “n choose k”) to calculate probabilities where you can get the same result in multiple ways.
So if you’re guessing the suit of a hidden playing card, you have a 1/4 chance of guessing correctly. If you guess for 10 different cards and get 5 correct, you might be tempted to say your feat was as likely as 
, and that you might have psychic powers. What you’re missing, though, is the number of ways you could have guessed correctly, 
, which means the probability of correctly guessing half the cards was 0.058. You could still be psychic, but now it doesn’t seem quite as likely.
Tune in later for a crash-course introduction to P-values and hypothesis testing, so we can conclude with more certainty one way or the other on your psychic abilities.
Tags: links, statistics