Please help me. A travel airline surveyed 950 business travelers 19% responded. What is the probability that more than 25% say that their travel was internal company visit

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Travel weekly International air transport association survey asked business travelers about their purpose for their most recent business, trip. 19% respond that it was for an internal company visit. Suppose950 business travelers are randomly selected. What is the probability that more than 25% of the business tracers say that their most recent business trip was an internal company visit

- queelius

This is probably a problem that can be solved more appropriately (and more effectively) using stats, but since I haven't taken stats in such a long time I don't feel inclined to solve it that way. I'm going to go with a pure probability (number of ways) approach -- I apologize to anyone reading this ahead of time.
Imagine a branching tree, where each node in the tree can go in one of two directions with probability p = 0.19 that they take the outbound edge for "most recent business trip was an internal company visit" and otherwise they take the other edge with probability (1-p) = 0.81. If we do this for 950 people, and n of them took the p edge, and (950 - n) took the (1-p) edge, then the probability of that path through the tree is p^n * (1-p)^(950-n). A very low probability, to be sure, no matter what n is. So, now we want to figure out all the ways in which a path of n edges are the "p" edges, which is just (950 choose n) ways.
So: (950 choose n) * [ p^n * (1-p)^(950-n) ]
If we allow n to be a non-whole number (thanks to various extensions to the factorial, this is possible), then we can choose n to be a fraction equal to 950 * 0.25 = 237.5.
So, the final equation is: (950 choose 237.5) * p^237.5 * (1-p)^(950 - 237.5)
This will be a small number. I'm not sure it makes much sense, either. I think a more reasonably interpretation is to say, ok, we can't have 25% on the nose -- a person can't be divided. So, find a close whole number approximation, e.g., 238 people. In fact, we also don't want 25% on the nose, but rather in the range of 24.5% to 25.5%, being sure to round to the nearest whole numbers for the end-points -> 233 to 242 people.
So, sum over this, e.g.: [sum from n = 233 to n = 242] (950 choose n) * [ p^n * (1-p)^(950-n) ]
I'm sure this is a terrible approach, and there are statistical approaches that will deal with this problem much more effectively.

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