A five-horse race is about to start. The probabilities that each horse will win are:
- Ajax: 40%
- Benji: 38%
- Cody: 18%
- Dusty: 3%
- Ember: 1%
Can you guess who will win?
There are several reasonable guesses you could make. For example, “Ajax” is a good guess, but “Ajax or Benji or Cody” is fine too. But some guesses, like “Cody or Ember,” are terrible.
What are the norms that govern guessing in this kind of context? Philosophers have become interested in that question recently (e.g., Holguín 2022, Dorst and Mandelkern 2022, Linnemann and Azhar 2025; our opening example is from Skipper 2023). It is a surprisingly rich question, because the answer does not obviously fall out of standard probability theory. For example, “Ajax or Ember” is a terrible guess, but the probability that either Ajax or Ember will win is higher than the probability that Ajax will win, and “Ajax” is a great guess.
With my colleagues Neil Bramley and Chris Lucas, I recently collected experimental data on how people guess. Our task was very simple. Participants looked at a box with colored balls, like this one:
Then we asked them to guess what color would come out if someone drew a ball at random. They could compose their guess by clicking on four buttons:
For example, to compose the guess “red or green,” you would click on “Red” and then “Green.” You could include any number of colors from one to four in your guess.
Here are the results! In this figure, each panel displays data for a different box. The numbers above the panel represent the proportion of colors – for example, “6 4 1 1” would correspond to the box shown above, with six red balls, four green balls, one blue, and one yellow ball.
We can see that for a box where all colors have equal proportions (3 3 3 3), almost all participants mention all colors (they guess, for example, “red or green or blue or yellow”). But for a box where one color dominates (9 1 1 1), most people only mention one color (for example, they guess “blue” if nine balls are blue). But between these two extremes, there is a lot of diversity in people’s guesses. For example, for the box “6 3 2 1,” about half of the participants mention one color, and half mention two colors.
Of course, the interesting question is whether theories of guessing proposed by philosophers can account for the data. We looked at an account by Kevin Dorst and Matt Mandelkern (2022). Abstracting from the mathematical details, their idea is that people want to make guesses that have a high probability of being true, but also do not mention too many possible outcomes. In other words, guessing is a trade-off between accuracy and specificity. The predictions from the theory are in green, alongside people’s data in white:
The theory fits the data pretty well.
Chris, Neil, and I also proposed another theory of how people might guess. Our idea is that a guess like “red or green” can be seen as implicitly encoding a probability distribution where red and green are both more probable outcomes than the other colors. And people make guesses that encode a distribution that is “close” to the actual distribution. So, if there are, for example, six red and four green balls in the box, the distribution encoded by “red and green” is close enough to the actual probability distribution that it is a good guess. The predictions from our theory are in purple:
The theory also gives a good account of the data. As you can see, the trade-off account and our account make fairly similar predictions. But there are some cases where they differ. For example, in the box “5 3 3 1,” we predict that people will either mention one color, or mention three colors. But the trade-off theory predicts that most people will mention two colors. Aligning with our prediction, people mostly mentioned either one or three colors. To see why this is an intuitive result, imagine a box with five red, three blue, and three yellow balls, as well as one green ball. It seems strange to guess “red or blue” in that context. According to our theory, this is because the guess “red or blue” encodes a distribution where blue is more likely than yellow, which isn’t the case here.
Of course, this is an active area of research, and other researchers might propose new theories of guessing in the future. Our data (freely available at https://osf.io/wz649/) give them a nice opportunity to see how their account compares with people’s intuitions.
For more details (and more experiments!), you can read our forthcoming paper at https://osf.io/preprints/psyarxiv/gy2fv_v3.
Literature
Dorst, Kevin, and Matthew Mandelkern (2022): “Good Guesses,” Philosophy and Phenomenological Research 105(3), 581–618. (Link)
Holguín, Ben (2022): “Thinking, Guessing, and Believing,” Philosophers’ Imprint 22, 6. (Link)
Linnemann, Niels, and Feraz Azhar (2025): “Better Guesses,” Philosophy and Phenomenological Research 110(2), 661–686. (Link)
Quillien, Tadeg, Neil Bramley, and Christopher G. Lucas (forthcoming): “Lossy Encoding of Distributions in Judgment Under Uncertainty,” Cognitive Psychology. (Link)
Skipper, Mattias (2023): “Good Guesses as Accuracy-Specificity Tradeoffs,” Philosophical Studies 180(7), 2025–2050. (Link)