An engineering firm consisting of 3 programmers and 6 circuit engineers. The emp
ID: 1814681 • Letter: A
Question
An engineering firm consisting of 3 programmers and 6 circuit engineers. The employees are randomly assign to 3 teams of 3, labeled Group 1, Group 2, and Group 3.
a) How many ways are there to assign the 9 employees to Groups 1, 2 & 3?
b) How many ways are there to assign the 9 employees to these groups so that every group has one programmer?
c) What is the probability that, after random assignment, every group includes one programmer?
d) What is the probability that, after random assignment, all three programmers are in a same group?
d) What is the probability that, after random assignment, exactly one group is left without programmer?
Please help, taking Random Signals and Systems, we have gone over total probability theorem, bayes rule, PMF...
Please show steps on how to derive answer for each part.Thank You
Explanation / Answer
a) total no of ways = no of ways to select 3 out of 9 for G1 * no of ways to select 3 out of remaining 6 gor G2* no of ways to select 3 out of remaining 3
= 9C3 * 6c3 * 3C3 = (9*8*7/1*2*3)* (6*5*4/1*2*3)*(1*2*3/!1*2*3) = 84*20*1 = 1680
b) no of ways to assign programmer to each group = 3! = 3*2*1 =6
no of ways to assign ce = 6C4* 4C2*2C2 = 15*6*1 = 90
total no ways so that every group has one programmer = 90*6 = 540
c) probability that, after random assignment, every group includes one programmer = total no ways so that every group has one programmer/total no of ways = 540/1680 = 0.321
d) no of ways to assign all three programmers are in a same group = no of ways to select a group in which programmer are there * no of ways to assign ce = 3*(6C3*3C3) = 3 * 60 *1 = 180
probability that, after random assignment, all three programmers are in a same group = no of ways to assign all three programmers are in a same group/ total no of ways= 180/1680 = 0.107
e)exactly one group is left without programmer = 1 - probability that, after random assignment, every group includes one programmer - programmers are in a same group/ total no of ways = 1-0.321-0.107 = 0.572