A committee consists of six men and four women .A subcommittee is made by randomly choosing three of the committee members. What is the probability that (a) they are all women (b) two of them are men?
(a) Well, since there are four women in the committee and three members are being randomly chosen for the subcommittee, we can use the concept of combination. The total number of ways to choose three members from the committee is C(10, 3). The number of ways to choose all women for the subcommittee is C(4, 3). Therefore, the probability that they are all women is:
P(all women) = C(4, 3) / C(10, 3)
Now, let's take a little journey through the delightful world of math:
C(4, 3) = 4! / (3! * (4-3)!) = 4
C(10, 3) = 10! / (3! * (10-3)!) = 120
So, the probability that they are all women is:
P(all women) = 4 / 120 = 1 / 30
(b) Similarly, the number of ways to choose two men from the six available is C(6, 2), and the number of ways to choose one woman from the four available is C(4, 1). Therefore, the probability that two of them are men is:
P(two men) = (C(6, 2) * C(4, 1)) / C(10, 3)
Now, hold onto your fun hat because here comes some mathematical magic:
C(6, 2) = 6! / (2! * (6-2)!) = 15
C(4, 1) = 4! / (1! * (4-1)!) = 4
C(10, 3) = 10! / (3! * (10-3)!) = 120
So, the probability that two of them are men is:
P(two men) = (15 * 4) / 120 = 60 / 120 = 1 / 2
Remember, math can be fun, especially when it involves clowns!
To find the probability, we need to determine the total number of possible subcommittees and the number of subcommittees that meet the given criteria.
(a) Probability that all three members are women:
To calculate this probability, we need to find the number of possible subcommittees consisting of three women and divide it by the total number of possible subcommittees.
Total number of subcommittees: This can be calculated using combinations. We can choose 3 members out of 4 women in "4 choose 3" ways.
Total number of subcommittees = C(4, 3) = 4
Number of subcommittees with all women: Since there are four women in the committee, we can choose all three members from the women in "4 choose 3" ways.
Number of subcommittees with all women = C(4, 3) = 4
Probability (a) = Number of subcommittees with all women / Total number of subcommittees = 4/4 = 1
Therefore, the probability that all three members of the subcommittee are women is 1.
(b) Probability that two of the members are men:
To calculate this probability, we need to find the number of possible subcommittees consisting of two men and one woman and divide it by the total number of possible subcommittees.
Total number of subcommittees: This can be calculated using combinations. We can choose 2 members out of 6 men in "6 choose 2" ways, and we can choose 1 member out of 4 women in "4 choose 1" ways.
Total number of subcommittees = C(6, 2) * C(4, 1) = 6 * 4 = 24
Number of subcommittees with two men and one woman: We can choose two members from the six men in "6 choose 2" ways and choose one member from the four women in "4 choose 1" ways.
Number of subcommittees with two men and one woman = C(6, 2) * C(4, 1) = 24
Probability (b) = Number of subcommittees with two men and one woman / Total number of subcommittees = 24/24 = 1
Therefore, the probability that two of the members of the subcommittee are men is 1.
To find the probability, we need to know the total number of possible outcomes as well as the number of favorable outcomes.
(a) Probability that all three members are women:
To calculate this, we need to determine the total number of possible subcommittees and the number of subcommittees consisting of all women.
Total number of possible subcommittees: This is the same as choosing any 3 members from the given 10 committee members, regardless of their gender. It can be calculated using the combination formula: C(10, 3) = 10! / (3! * (10 - 3)!) = 120.
Number of subcommittees consisting of all women: Since there are 4 women in the committee, we need to choose all 3 of them. This can be calculated using the combination formula: C(4, 3) = 4! / (3! * (4 - 3)!) = 4.
Therefore, the probability that all three members are women is 4/120 = 1/30.
(b) Probability that two of the three members are men:
To calculate this, we need to determine the total number of possible subcommittees and the number of subcommittees consisting of two men.
Total number of possible subcommittees: This is the same as choosing any 3 members from the given 10 committee members, regardless of their gender. We already determined this to be 120.
Number of subcommittees consisting of two men: We can choose 2 men from the 6 men available, and then choose any 1 member from the remaining 4 women. This can be calculated using the product of combinations: C(6, 2) * C(4, 1) = (6! / (2! * (6 - 2)!)) * (4! / (1! * (4 - 1)!)) = 15 * 4 = 60.
Therefore, the probability that two of the three members are men is 60/120 = 1/2.
(a) 4/10 * 3/9 * 2/8
(b) 4/10 * 6/10 * 6/10 is the chance that only the 1st one chosen is a woman. Now, what about other options?