Assume last year I commuted to work in a taxi and suppose if there are 'n' taxis in th fleet I used and if I took one of these taxis every trip at random and with replacement then what is the number of trips, I needed to make in order to use every taxi in the fleet?
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This is the coupon collector's problem. 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. Then there is the issue of what "at random" means: some taxis may usually work when you are not commuting or in other parts of the city. |
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