Question 426302
<font face="Times New Roman" size="+2">


If you consider chosing a Democrat for any one of the positions as a success (nothing to do with politics, it is simply a way of looking at the problem), then you want the probability of 4 successes in 4 trials where the probability of success on any given trial is 0.6.  I say 0.6 because there are 10 people to choose from and 6 are considered a success.


Alternatively, if your conservative politics simply won't allow you to consider selecting a Democrat as a success, you could calculate the probability of 4 failures (failure to pick a Republican) in 4 trials where the probability of success is 0.4.  You will find that it reduces to the exact same calculation.


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\)p^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)!}]


So, for this problem you could either compute:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P_4(4,0.6)\ =\ \left(4\cr 4\right\)\left(0.6\right)^4\left(0.4\right)^{0}]


or


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P_4(0,0.4)\ =\ \left(4\cr 0\right\)\left(0.4\right)^0\left(0.6\right)^{4}]


And remembering that *[tex \Large \left(n\cr n\right\)\ =\ \left(n\cr 0\right\)\ =\ 1] and  *[tex \Large x^0\ =\ 1],


So either way, it reduces to


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P_4(4,0.6)\ =\ P_4(0,0.4)\ =\ \left(0.6\right)^{4}]


You can do your own arithmetic.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
<div style="text-align:center"><a href="http://outcampaign.org/" target="_blank"><img src="http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png" border="0" alt="The Out Campaign: Scarlet Letter of Atheism" width="143" height="122" /></a></div>
</font>