In late January, Daniel Litt posed an innocent probability puzzle on the social media platform X (formerly known as Twitter) — and set a corner of the Twitterverse on fire.
Imagine, he wrote, that you have an urn filled with 100 balls, some red and some green. You can’t see inside; all you know is that someone determined the number of red balls by picking a number between zero and 100 from a hat. You reach into the urn and pull out a ball. It’s red. If you now pull out a second ball, is it more likely to be red or green (or are the two colors equally likely)?
Of the tens of thousands of people who voted on an answer to Litt’s problem, only about 22% chose correctly. (We’ll reveal the solution below, in case you want to think it over first.) In the months since, Litt, a mathematician at the University of Toronto, has continued to confound Twitter users with a series of probability puzzles about urns and coin tosses.
His posts have prompted lively online discussions among research mathematicians, computer scientists and economists — as well as philosophers, financiers, sports analysts and anonymous fans. Some joked that the puzzles were distracting them from their real work — “actively slowing down economic research,” as one economist put it. Others have posted papers exploring the puzzles’ mathematical ramifications.
Litt’s online project doesn’t just highlight the enduring allure of brainteasers. It also demonstrates the limits of our mathematical intuition, and the counterintuitive nature of probabilistic reasoning. As Litt wrote, there’s “nothing more exhilarating than posing a multiple-choice problem on which 50,000 people do substantially worse than random chance.”
Quanta spoke with Litt about what makes a great puzzle, and why simple probability questions can be so deceptively difficult. (Warning: spoilers below!) The interview has been condensed and edited for clarity.
Your background is at the intersection of algebraic geometry and number theory. What made you start posting probability puzzles?
Probability theory is extremely far from what I usually think about mathematically. But I went to a bunch of probability talks, and I could really enjoy them despite being a total amateur, because they were about things that are reasonably familiar.
I started posing some simple probability questions to myself, and when the answers I found were cool or counterintuitive or surprising, I turned them into puzzles on Twitter. Already with flipping a coin, it’s easy to come up with questions of that nature.
Let’s talk urns. In your puzzle, once you pick a red ball, is the next ball more likely to be red or green?
Red.
Why is that?
My favorite way to think about this is due to George Lowther, a respondent who came up with the following explanation on X. He said, imagine that instead of starting with 100 balls, you start with 101 balls in a row. Pick a ball at random. Then color the balls to the left of it green and the ones to the right of it red. Throw that ball away, leaving 100 balls.
Then pick a second ball at random. That ball corresponds to the first ball in the original problem. The problem tells you that you picked a red ball, so it was to the right of the ball you threw away. Now pick a third ball. This ball is either to the left of the first ball, between the first ball and the second, or to the right of the second. In two of the three possibilities, the third ball is red. So the probability that the ball is red is 2/3.
Another respondent had a nice heuristic, which was that if you go fishing and quickly catch a fish, you should expect there to be a lot of fish in that lake. Similarly, if you’ve gotten one red ball already, that suggests that there are many red balls in the urn.
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