Question 393208
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The probability of anything is the number of ways that it can happen that you would consider a success divided by the number of ways it can happen total -- successes and failures.


There are 52 cards in the deck.  4 of them are 9s.


12 of them are face cards.


40 of them are not face cards.  (or just subtract the answer to the previous question from 1)


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 4 hearts out of 4 cards dealt:  *[tex \Large n\ =\ 4\ \ ], *[tex \Large k\ =\ 4], and *[tex \Large p\ =\ \frac{1}{4}]


So for 4 face cards out of 4 cards dealt:  *[tex \Large n\ =\ 4\ \ ], *[tex \Large k\ =\ 4], and *[tex \Large p\ =\ \frac{3}{13}]


And for 4 even numbered cards out of 4 cards dealt:  *[tex \Large n\ =\ 4\ \ ], *[tex \Large k\ =\ 4], and *[tex \Large p\ =\ \frac{6}{13}]


You get to do your own arithmetic.  Hint: *[tex \LARGE \left(n\cr n\right\)\ =\ 1\ \forall\ n\ >\ 0\ \in\ \mathbb{Z}\ ] and *[tex \LARGE x^0\ =\ 1\ \forall\ x\ \in\ \mathbb{R}] 


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