Expected 2's

What this Bayesian inference and expected value question tests

This is a medium-difficulty probability problem that combines Bayesian reasoning with conditional expectation. It appears frequently in quant trading interviews because it requires you to update your beliefs about the state of the world (which urn you drew from) based on observed evidence, then compute an expected value under that updated belief.

The core challenge is recognizing that your first draw—a 2—tells you something about which urn you likely selected. You must use Bayes' theorem to calculate the posterior probability of each urn, then compute the expected number of 2's in the remaining 40 draws by weighting the draw probabilities for each urn by their posterior likelihoods. This tests both your probability mechanics and your intuition about how evidence shifts belief.

• Bayes' theorem and posterior probability
• Conditional expectation and the law of total expectation
• Independence and sampling with replacement