\underline{1}

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 ≤ X ≤ 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 · 5 · 4}{10 · 9 · 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} ·_{6}C_{2}}{_{10}C_{3}} = \frac{4 · 6 · 5}{2 · 1} · \frac{3 · 2 · 1}{10 · 9 · 8} = \frac{1}{2}

Similarly,

\begin{matrix}X & 0 & 1 & 2 & 3 \\ P (X = i) & \frac{1}{6} & \frac{1}{2} & \frac{6}{20} & \frac{1}{30} \end{matrix}

P_{r}(1 ≤ X ≤ 3) = 1 − P_{r}(X = 0) = 1 − \frac{1}{6} = \frac{5}{6}