Probability puzzles, one good question at a time.
Think in Odds is a free daily probability puzzle. Spend a few minutes making a prediction, check your answer, and work through the explanation. No account is required. New editions arrive at 8 p.m. Eastern.
How to solve a probability puzzle
- Define the experiment. What is random? Which outcomes can happen? Read the assumptions before you count.
- Check whether outcomes are equally likely. For a fair die, each face has probability 1/6. For a biased die, counting faces alone is not enough; add their probability weights.
- Keep only the outcomes consistent with the evidence. Conditional probability is P(A given B) = P(A and B) / P(B), when P(B) is positive. The denominator describes the world after learning B.
- Use independence only when it applies. Multiplying probabilities is appropriate for independent events. Otherwise, multiply by the conditional probability of each next step.
- Check the scale. A probability lies between 0 and 1. Expected values can exceed 1, and need not be possible individual outcomes.
A small example: two fair dice
Roll two independent fair six-sided dice. What is the probability that their sum is 7? There are 36 equally likely ordered outcomes. Six work: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). The probability is 6/36 = 1/6.
The common trap is to count the possible sums from 2 to 12 as equally likely. They are not: a sum of 2 has just one ordered outcome, while a sum of 7 has six.
Start with a released puzzle
These puzzles cover counting, evidence, expectation, and the assumptions that make an answer meaningful. Hints are optional; solutions stay hidden until you choose to reveal them.
- Two jars, one lucky draw — Strategy, about 6 minutes
- How long until a six? — Expected value, about 3 minutes
- One green ball changes the odds — Bayesian reasoning, about 6 minutes
- A boy born on Tuesday — Conditional probability, about 6 minutes
- Put the champion in the middle? — Strategy, about 5 minutes
- Heads, then tails — Expected value, about 4 minutes
- The higher of two dice — Expected value, about 5 minutes
- The coin with two heads — Bayesian reasoning, about 4 minutes
What happens after I answer?
Correct answers show a compact result. Reveal the worked solution to compare your reasoning with the explanation and its illustration. Your progress is saved in this browser. A shared challenge links to the puzzle without exposing the answer.
Can I use this for interview practice?
These puzzles build habits useful in quantitative interviews: stating assumptions, distinguishing independent from dependent events, and explaining why a calculation fits the problem. They are practice for intuition, not a complete interview syllabus or a score predicting interview performance.
Who makes Think in Odds?
David Z. Chen maintains Think in Odds. The first 30 daily editions are curated; later editions revisit mathematical families with spaced numerical practice. If a question seems ambiguous or an explanation looks wrong, send feedback.