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{\bf Question}

Suppose that a box contains 7 red balls and 3 blue balls.  5 balls
are selected at random, without replacement.  Let $X$ denote the
number of red balls that will be obtained. Determine the pmf of
$X$ and sketch it.


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{\bf Answer}

The possible values of $X$ are 2, 3, 4 and 5. By considering
possible combinations of the five balls selected, it can be seen
that

$$P\{X=k\}=\frac{\left(\begin{array}{c}7\\k\end{array}\right)
\left(\begin{array}{c}3\\5-k\end{array}\right)}
{\left(\begin{array}{c}10\\5\end{array}\right)},\ \ \ \
k=2,3,4,5$$

and so $$\displaystyle P\{X=2\}=\frac{1}{12},\
P\{X=3\}=\frac{5}{12},\ P\{X=4\}=\frac{5}{12},\
P\{X=5\}=\frac{1}{12}.$$


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