A confusing probability question: Red and green balls in an urn

Original Article ↗ Hacker News Discussion ↗

A probability puzzle about drawing red and green balls from an urn highlights how counterintuitive Bayesian reasoning can be. When the number of red balls is chosen uniformly at random and the first draw is red, many people expect the next draw to favor green because one red has been removed, but the correct result is that red is actually more likely (2/3) once you account for how that first red updates the distribution of possible urn compositions. Commenters explore why simulations often go wrong, how different prior distributions (uniform vs. binomial or normal) change the answer, and how ambiguous wording like “more likely” contributes to widespread misunderstanding, much like in the Monty Hall and “boy or girl” paradoxes.

Core problem and headline result

  • Urn has 100 balls; unknown number of reds n is chosen uniformly from 0–100. You draw one ball and it’s red. Question: is the next draw more likely red, green, or equal?
  • Consensus among people who worked it through: the next ball is more likely red, with probability 2/3 (independent of the total ball count, as long as ≥2).
  • Intuition: seeing red first makes it more likely you started from a “red-heavy” urn than a “red-scarce” one, and that effect dominates the fact you removed one red.

Bayesian / configuration perspectives

  • Many explanations frame it as: initially each n is equally likely; after seeing a red, posterior weight shifts toward larger n.
  • Explicit formula: posterior P(n | first red) ∝ n, yielding a heavily skewed distribution toward red-heavy urns.
  • Several posts count “worlds” or configurations (over n and draw positions) and show that, conditioned on a first red, red-red sequences occur twice as often as red-green.

Language and ambiguity

  • Substantial debate about wording: “more likely” could mean:
    • more likely red than green on the second draw, or
    • more likely red on the second draw than the first draw was to be red.
  • Some argue the poll’s phrasing invites misinterpretation; others say the intended reading is clear: “Which color is more likely next?”

Alternative priors and distributions

  • Key point: the uniform prior over n drives the counterintuitive result.
  • If balls were independently colored by a fair coin (binomial prior), observing one red gives no information about the next draw; probability stays 1/2.
  • With a normal-like prior concentrated near 50–50, the update from one red is tiny; result is close to 1/2, matching “common sense.”

Simulations and common mistakes

  • Multiple commenters use Monte Carlo simulations; some initially implemented a binomial urn by mistake, getting ~50/50 and correcting later.
  • Others show how mis-handling impossible cases (e.g., n=0 after drawing red) or trial counting can silently bias simulated probabilities.

Related problems and pedagogy

  • Follow-up: drawing 99 reds in a row implies the last ball is overwhelmingly likely red, not 50/50, since the “all-red” urn explains the data far better than “one green.”
  • Comparisons are drawn to Monty Hall and “boy/girl” paradoxes: small bits of information, conditioned properly, can drastically change probabilities.
  • Several see the question as a useful interview or teaching tool to expose intuitions, reasoning style, and understanding of conditioning.