In how many ways can a committee of three be chosen from 10 female and 20 male? if
a) All are equally eligible?
b) One female?
c) All are female?
d) Two female?
e) What is probability of selecting one female in a committee?
f) What is probability of selecting two female in a committee?
g) What is probability of selecting three female in a committee?
a) To choose a committee of three from 10 female and 20 male, we can use the combination formula. The number of ways to choose a committee of three from 10 female is denoted as C(10, 3), which can be calculated as:
C(10, 3) = 10! / (3!(10-3)!) = 10! / (3!7!) = (10*9*8) / (3*2*1) = 120
Similarly, the number of ways to choose a committee of three from 20 male is C(20, 3) = 20! / (3!(20-3)!) = 1140.
So, the total number of ways to choose a committee of three, regardless of gender, is:
C(10, 3) * C(20, 3) = 120 * 1140 = 136,800.
b) To choose a committee of three with exactly one female, we need to choose one female from the 10 available and two males from the 20 available. The number of ways to do this is:
C(10, 1) * C(20, 2) = 10 * 190 = 1,900.
c) To choose a committee of three with all females, we simply need to choose three females from the 10 available. The number of ways to do this is:
C(10, 3) = 120.
d) To choose a committee of three with exactly two females, we need to choose two females from the 10 available and one male from the 20 available. The number of ways to do this is:
C(10, 2) * C(20, 1) = 45 * 20 = 900.
e) The probability of selecting one female in a committee is calculated by dividing the number of ways to choose a committee with exactly one female (1,900) by the total number of ways to choose a committee of three, regardless of gender (136,800):
P(one female) = 1,900 / 136,800 ≈ 0.0139 or 1.39%.
f) The probability of selecting two females in a committee is calculated by dividing the number of ways to choose a committee with exactly two females (900) by the total number of ways to choose a committee of three, regardless of gender (136,800):
P(two females) = 900 / 136,800 ≈ 0.0066 or 0.66%.
g) The probability of selecting three females in a committee is calculated by dividing the number of ways to choose a committee with all females (120) by the total number of ways to choose a committee of three, regardless of gender (136,800):
P(three females) = 120 / 136,800 ≈ 0.0009 or 0.09%.