Exactly one of six similar keys opens a certain door. If you try the keys, one after another, what is the expected number of keys that you will have to try before success?

• 2.5
• 3
• 3.5
• 4
• 4.5

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Solution: C

The probability that you need to try [math]k [/math] keys equals

[[math]] \frac{5}{6} \frac{4}{5} \cdots \frac{6-k-1}{6-k} \frac{1}{6-k-1} = \frac{1}{6}. [[/math]]

Hence the probability distribution is uniform on [math]\{1,\ldots,6\}[/math]. Then the expected value equals

[[math]] \frac{1}{6}\sum_{k=1}^6 k = 3.5. [[/math]]