Collect Them All

What this coupon-collector expected-value question tests

This is an easy probability question that appears frequently in quant interviews because it teaches a clean, generalizable approach to sampling-with-replacement problems. You are asked to find the expected number of rolls needed to observe every distinct outcome of a fair die.

The key insight is to decompose the problem into stages: how many rolls until you see the first new face, how many more until you see a second new face, and so on. This transforms a tricky global question into a sum of simpler conditional expectations. The reasoning is structural rather than computational, and the final answer is a clean fraction.

• Linearity of expectation across stages
• Geometric distributions (waiting time for a success at a given probability)
• The coupon-collector problem as a prototype