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Thread: Complex probability relating to NYC taxi cabs

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    Complex probability relating to NYC taxi cabs




    There are 13,000 taxi cabs in New York City.

    Let's say I have taken 600 rides (each one in a random cab).

    What are the chances that I have ridden in at least one cab three times?

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    Re: Complex probability relating to NYC taxi cabs

    In general you will need the multinomial distribution.

    Let X_i be the number of times you have ridden in for the i-th cab, i = 1, 2, \ldots, 13000.

    Then (X_1, X_2, \ldots, X_{13000}) \sim \text{Multinomial}\left(600; \frac {1} {13000}, \frac {1} {13000}, \ldots, \frac {1} {13000}\right)

    You want to calculate

    \Pr\bigcup_{i=1}^{13000} \{X_i \geq 3\}

    Now you have two ways to count this. First is to use De Morgan's law:

    \Pr\bigcup_{i=1}^{13000} \{X_i \geq 3\} = 1 - \Pr\bigcap_{i=1}^{13000} \{X_i < 3\}

    Then for the latter probability, you know you just need to consider X_i \in \{0, 1, 2\} only.

    To enumerate that, note that in (X_1, X_2, \ldots, X_{13000}), there must be

    k 2's, 600 - 2k 1's and 12400 + k 0's, k = 0, 1, 2, \ldots, 300

    Therefore we have:

    \Pr\bigcap_{i=1}^{13000} \{X_i < 3\} = \sum_{k=0}^{300} \frac {13000!} {k!(600 - 2k)!(12400+k)!} \frac {600!} {(2!)^k (1!)^{600-2k}(0!)^{12400+k}} \frac {1} {13000^{600}}

    and can calculate accordingly.

    Second way is to use inclusive-exclusive principle, which allow you to obtain an alternating series, and you may truncate the series as an upper/lower bound when you have calculate enough precision.

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    bugman (06-05-2013), TheEcologist (06-05-2013), trinker (06-05-2013)

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    Re: Complex probability relating to NYC taxi cabs

    The depth of your knowledge BGM, never stops to amaze me.
    The true ideals of great philosophies always seem to get lost somewhere along the road..

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    Re: Complex probability relating to NYC taxi cabs

    lol I have not verify the formula yet. Not sure if that is correctly deduced.

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    Re: Complex probability relating to NYC taxi cabs

    Thanks for replying.

    Can you please tell me what the final odds are?

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    Re: Complex probability relating to NYC taxi cabs


    The sought probability can be very accurately approximated by the Poisson probability 1-e^(-0.211953)=0.1910 where 0.2119538... is obtained from bin(600,3)/13,000^2.

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