Question 325274
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The probability of *[tex \Large k] successes in *[tex \Large n] trials where the probability of success on any given trial is *[tex \Large p] 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 computed using *[tex \Large \frac{n!}{k!(n\,-\,k)!]


For this problem, given a fair coin, the probability of success on any given trial is *[tex \Large \frac{1}{2}], and you want 3 successes out of 3 trials, hence:


*[tex \LARGE\ \ \ \ \ \ \ \ \ \ P_3\left(3,\frac{1}{2}\right)\ =\ \left(3\cr{3}\right)\left(\frac{1}{2}\right)^3\left(\frac{1}{2}\right)^{0}]


You can calculate this for yourself.  Hint:  *[tex \Large \left(n\cr{n}\right)\ =\ 1] and anything to the zero power is 1.


That gives you the probability of winning.  The probability of losing is *[tex \Large 1\ -\ p_w].


The expected outcome is the amount you lose whenever you lose times the probability of losing plus the amount you win whenever you win times the probability of winning.



John
*[tex \Large e^{i\pi}\ +\ 1\ =\ 0]
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