Solution
Distinct Faces
Show the problem again
A fair die is rolled six times. What is the expected number of distinct faces that appear? Give an exact expression or a decimal rounded to 2 decimal places.
Worked solution
The answer is 6[1 − (5/6)⁶] = 31031/7776 ≈ 3.99. For each face i, let X_i be the indicator that face i appears at least once; P(X_i = 1) = 1 − (5/6)⁶. By linearity of expectation, E[distinct faces] = 6[1 − (5/6)⁶] ≈ 3.99. Linearity works regardless of the dependence between indicators.
Source: Standard dice-counting exercise from introductory probability texts. Statement written for AxiomIQ.