Question 575462
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The probability of *[tex \Large k] successes in *[tex \Large n] trials where *[tex \Large p] is the probability of success on any given trial is given by:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P_n(k,p)\ =\ \left(n\cr k\right\)\left(p\right)^k\left(1\,-\,p\right)^{n\,-\,k}]


Where *[tex \LARGE \left(n\cr k\right\)] is the number of combinations of *[tex \Large n] things taken *[tex \Large k] at a time and is calculated by *[tex \Large \frac{n!}{k!(n\,-\,k)!}]


You need the probability of 0 plus 1 plus 2 plus ... plus 11 plus 12 out of 250 trials at a probability of 0.08 for success on any given trial.  Hence:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P_{250}(<12,0.08)\ =\ \sum_{i\,=\,0}^{12}\left(250\cr\ \ i\right\)\left(0.08\right)^i\left(0.92\right)^{250\,-\,i}]


Now I don't care who you are, that is some pretty ugly arithmetic.  If you are a really big glutton for punishment, have at it.  But here is a much easier way:


Open an Excel spreadsheet (or Numbers if you are using a Mac).  Enter the following in any convenient empty cell:


=BINOMDIST(12,250,0.08,true)


And hit "Enter"


3 decimal places is probably way more precision than you need.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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