Three friends covering all three mealtimes

Question

Three friends each independently pick breakfast, lunch or dinner at random, each with probability $\tfrac13$ , as the time to visit a restaurant. What is the probability that every mealtime is chosen by someone?

Short answer

$2/9$ . All three times are covered only if the choices are all different: $3!=6$ of the $3^3=27$ equally likely outcomes.

Adapted from OpenQuant; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Counting and Combinatorics
Independence
Sample Spaces, Events, and the Probability Axioms

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