Question 5.16: The student population of a certain college in Massachusetts......
The student population of a certain college in Massachusetts has been found to be made up as follows: 50% are from Massachusetts (or in-state students), 20% are from the other states of the United States (or out-of-state students), and 30% are from other countries (or foreign students). If 10 students from the college are randomly selected by a company conducting a survey, what is the probability that 6 of them will be in-state students, 2 are out-of-state students, and 2 are foreign students?
Let K_1 be the number of in-state students among the 10 selected students, K_2 the number of out-ofstate studentsamong the 10 selected students, and K_3 the number of foreign studentsamong the 10 selected students. Similarly, let p_1 denote the probability that an in-state student is randomly selected, p_2 the probability that an out-of-state student is randomly selected, and p_3 the probability that a foreign student is randomly selected. Since \,\displaystyle p_1=0.5,p_2=0.2 and p_3=0.3, the desired result is
\,\displaystyle p_{K_1k_2K_3}\left (6,2,2\right )=\binom{10}{622}\left (0.5\right )^6\left (0.2\right )^2\left (0.3\right )^2=\frac{10 !}{6 ! 2 ! 2 !}\left (0.5\right )^6\left (0.2\right )^2\left (0.3\right )^2\\ \,\displaystyle =1260\left (0.5\right )^6\left (0.2\right )^2\left (0.3\right )^2=0.0709