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How many taxis are there?

Question

I arrive at a small town. Leaving the airport i see five taxis out front. The city sponsors the taxis. Each taxicab has a unique number on it. The largest number on the taxicab was 90. How many taxis are there in the city?

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It's the German Tank Problem:

N~=m-1+m/k, where N is total taxis, m is highest number, and k is number of taxis spotted.

N~=90-1+90/5=107 taxis.

Based on probability mass function, approach gives expected value of

N=(m-1)*(k-1)/(k-2)
N=89*4/3=119.

http://en.wikipedia.org/wiki/German_tank_problem#Example
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uhm... 86? lol. I have no idea.

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I appreciate the numerical theory in the answer, but I don't buy into that. A Monte Carlo run shows that 90 is the most likely value, though this decreases slowly. Picking 5 random values from a list, where N is the highest value in the original list, and 90 is the highest of the 5 observed values, and running for 1 million trials, I get a table that looks like this. (I've not listed all values to 190, the last N that had any hits.)

N *** % 90 4.63% 91 9.01% 92 13.13% 93 17.02% 94 20.74% 95 24.21% 96 27.53% 97 30.72% 98 33.71% 99 36.58% 100 39.30%

From this run, the chances are even that N is less than or more than 105.

How to reconcile this result with the numerical analysis?

Also, if the numerical analysis has 2 valid answers, then there must be a range of valid answers in that region. It seems to me that this problem doesn't have an answer, but a handwaving "something about here" result.

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I marked the post as being correct. Not necessarily both answers. His "frequentest approach" was the one i was looking for.

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