Example One bag contains 4 red balls and 6 white balls. When taking out three spheres at the same time, find the random variable $X$ and the probability distribution $f$ that represent the number of red balls and draw the graph. Alos find $P(X = 1), P(1 \leq X \leq 3)$．

Answer The combination of taking out 10 to 3 pieces is ${}_{10} C_{3} = 120$. Also, The fact that red is zero out of three means that white is the same as three, so three are taken out of six whites, and the combination is ${}_6 C_{3} = 20$ ways．Thus, let $X$ be the number of red balls. Then

$$P_{r}(x = 0) = \frac{{}_6 C_{3}}{{}_{10} C_{3}} = \frac{6\cdot 5 \cdot 4}{10\cdot 9 \cdot 8} = \frac{1}{6}$$

One out of three is red means that the other two are white

$$P_{r}(X = 1) = \frac{{}_4 C_{1} \cdot {}_6 C_{2}}{{}_{10} C_{3}} ... ... 6 \cdot 5}{2\cdot 1}\cdot \frac{3\cdot2\cdot1}{10\cdot 9\cdot8} = \frac{1}{2}$$

Similarly,

$$\begin{array}{\vert c\vert cccc\vert} \hline X & 0 & 1 & 2 & ... ...frac{1}{2} & \frac{6}{20} & \frac{1}{30} \\ \hline \end{array}$$

$$P_{r}(1 \leq X \leq 3) = 1 - P_{r}(X = 0) = 1 - \frac{1}{6} = \frac{5}{6}$$