Prize door found with a wheel that sometimes tells

Question

A game show has three doors, exactly one of which hides a $1 million prize. Before choosing, you spin a wheel: with probability $\tfrac35$ it reveals the prize door, and otherwise it gives no information and you pick one of the three doors uniformly at random. What is the probability that you find the prize?

Short answer

$\tfrac{11}{15}$ . Condition on the wheel: with probability $\tfrac35$ you win for sure, and with probability $\tfrac25$ you have a one-in-three guess. Total: $\tfrac35+\tfrac25\cdot\tfrac13=\tfrac{11}{15}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating
Continuous Random Variables and Probability Density Functions

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