My August diary included this:
Roll three normal dice. What is the probability of "getting a three"? That is, what's the chance that the numbers that came up made a three in some combination: (1, 1, 1), say, or (1, 2, 4), or (1, 3, 2), or (5, 3, 1)? As opposed to numbers that don't, like (1, 4, 1), (2, 2, 2), or (6, 5, 2)?
This should be straightforward. There are 216 equal-probability results of the throw. So you just have to count how many of those possibilities will "get" you a three, then divide that number by 216.
Yet for some reason it's hard to get the answer right. The 16th-century genius and gambler Girolamo Cardano, who wrote the first book-length study of probability theory (and who I covered in Chapter 4 of Unknown Quantity), got it wrong. He also got the wrong answers for getting a four, a five, and a six on a roll of three dice.
Stephen Stigler, Professor of Statistics at the University of Chicago, gave this problem to his students two years running. He reports that only a third of them found the correct answer for "getting a three"; for "getting a four," less than a quarter of students got the right answer.
I myself had a go at the probabilities for getting a one, a two, a three, a four, a five, or a six. I set up a spreadsheet with 216 lines, one for each equal-probability outcome of the throw. Then I eyeballed through, marking up each outcome that made the number I sought.
I got the right answers for ones, threes, and fours, but not for twos, fives, or sixes (though I did better than Cardano). What's up with this?