Question 2. (25 points) In a certain department, there are F female and M male f
ID: 3306626 • Letter: Q
Question
Question 2. (25 points) In a certain department, there are F female and M male faculty members; and you can assume that F 23,M 23. A 4-person committee is to be created by random selection of the faculty members. a) How many possible distinct committees are there? b) What is the probability that a committee has an equal number of male and female faculty members? c) How many distinct committees are there if there must be at least one male and at least one female faculty member on the committee. d) For the selection procedure of part c), what is the probability that a committee has an equal number of males and female faculty members? e) Evaluate parts b) and d) for F-M-3. Is the result surprising? Would you expect a similar result if the department added one new female and one new male faculty member?Explanation / Answer
a) Since there are F + M members and the committee is to be formed consisting of 4 members the number of committies = (F+M)C4.
b) If the committee has equal number of males and females, there should be 2 males and 2 females and the number of such committies = FC2*MC2.
c) If committees are to be formed with males only, there are MC4 ways.
If committees are to be formed with females only, there are FC4 ways.
=> Number of committees with atleast one male and atleast one female = (F+M)C4 - FC4 - MC4 (In case F and M are 3, the result is simply (F+M)C4.
d) Again, the committee can have equal number of males and females only if the number of each is 2. This also covers the criterion in c) The number is the same i.e FC2*MC2.
e) When F = M = 3
a) evaluates to 6C4 = 15.
b) evaluates to 3C2 * 3C2 = 3 * 3 = 9.
c) evaluates to 15.
d) also evaluares to 3C2 * 3C2 = 3 * 3 = 9.
The results are not very surprising except for c) which is a special case. In case the number is increased by one each a), b) and d) will follow similar pattern but c) will have a new case.
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