Expected items until the first defective one

Question

A production line turns out items one at a time. Each item is independently defective with probability $\tfrac15$ . An inspector examines them in order and stops at the first defective item.

What is the expected number of items inspected, including the defective one?

Accepted Answer

Let E be the expected count. The first item is always inspected. With probability 	frac15 it is defective and the inspector stops; with probability 	frac45 it is fine and the situation resets: E = 1 + 	frac45 E \quad\Longrightarrow\quad 	frac15 E = 1 \quad\Longrightarrow\quad E = 5. Takeaway. A one-in-five event arrives after 1/p = 5 trials on average.

Expected items until the first defective one

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