# Thread: Complex probability relating to NYC taxi cabs

1. ## 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?

2. ## Re: Complex probability relating to NYC taxi cabs

In general you will need the multinomial distribution.

Let be the number of times you have ridden in for the -th cab, .

Then

You want to calculate

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

Then for the latter probability, you know you just need to consider only.

To enumerate that, note that in , there must be

2's, 1's and 0's,

Therefore we have:

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.

3. ## The Following 3 Users Say Thank You to BGM For This Useful Post:

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

4. ## Re: Complex probability relating to NYC taxi cabs

The depth of your knowledge BGM, never stops to amaze me.

5. ## Re: Complex probability relating to NYC taxi cabs

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

6. ## Re: Complex probability relating to NYC taxi cabs

Can you please tell me what the final odds are?

7. ## 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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