# Probability associated with experiencing all outcomes

Assume last year I commuted to work in a taxi, and suppose if there are $n$ taxis in the fleet I used. If I took one of these taxis every trip at random and with replacement, then what is the number of trips I need to make in order to use every taxi in the fleet?

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The expected number is $$n \sum_{i=1}^n \frac{1}{i}$$ which is about $n(\log_e(n)+ \gamma) +\tfrac12$ where $\gamma \approx 0.577\ldots$ is the Euler–Mascheroni constant. But there is a wide dispersion and you will never be absolutely sure.