Meter needing four genuine quarters in a row

Question

A parking meter needs $1 in quarters, but any counterfeit quarter you insert resets the meter to zero. Each coin you pull from your (bottomless) purse is counterfeit with probability 0.4, independently. What is the expected number of coins you insert before the meter is paid?

Short answer

About $16.79$ . You need four consecutive genuine coins ( $p=0.6$ ). The expected wait for $r$ successes in a row is $\dfrac{1-p^r}{(1-p)p^r}$ ; with $p^4=0.1296$ this is $\dfrac{0.8704}{0.05184}\approx16.79$ .

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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