A 4-person leadership committee is randomly chosen from a group of 24 candidates. Ten of the candidates are men, and 14 are women.
What is the probability that the committee is all male or all female?
What is the probability that the committe has at least 1 man or at least 1 women?
All male probability: 10/24*9/23*8/22*7/21 = 5040/255,024
= 0.01976
All female probability:
14/24*13/23*12/22*11/21= 24,024/255,024
= 0.09420
prob. of at least one man and one woman:
= 1 - (prob. of zero males) - (prob. of zero females) = 1- 0.01976 - 0.09420
= 0.88604
To find the probability of an event, we need to determine the total number of possible outcomes and the number of favorable outcomes.
1. Probability that the committee is all male or all female:
To calculate this, we need to find the number of combinations where the committee is all male or all female and divide it by the total number of possible combinations.
Total number of possible combinations = C(24, 4)
= 24! / (4! * (24-4)!)
= 24! / (4! * 20!)
= (24 * 23 * 22 * 21) / (4 * 3 * 2 * 1)
= 10,626
Number of combinations with all male = C(10, 4)
= 10! / (4! * (10-4)!)
= 10! / (4! * 6!)
= (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)
= 210
Number of combinations with all female = C(14, 4)
= 14! / (4! * (14-4)!)
= 14! / (4! * 10!)
= (14 * 13 * 12 * 11) / (4 * 3 * 2 * 1)
= 1,001
Number of favorable outcomes = Number of combinations with all male + Number of combinations with all female
= 210 + 1,001
= 1,211
Probability = Number of favorable outcomes / Total number of possible combinations
= 1,211 / 10,626
= 0.114 or 11.4%
Therefore, the probability that the committee is all male or all female is 11.4%.
2. Probability that the committee has at least 1 man or at least 1 woman:
To calculate this probability, we can find the probability of the complement event first, which is the event that the committee consists of only one gender. Then, we can subtract this probability from 1 to find the desired probability.
Probability of the complement event (committee consists of only one gender):
Total number of possible combinations for a committee of all men = C(10, 4) = 210
Total number of possible combinations for a committee of all women = C(14, 4) = 1,001
Number of favorable outcomes (committee consists of only one gender) = Number of combinations with all men + Number of combinations with all women
= 210 + 1,001
= 1,211
Probability of the complement event = Number of favorable outcomes (committee consists of only one gender) / Total number of possible combinations
= 1,211 / 10,626
= 0.114 or 11.4%
Probability of the desired event (committee has at least 1 man or at least 1 woman) = 1 - Probability of the complement event
= 1 - 0.114
= 0.886 or 88.6%
Therefore, the probability that the committee has at least 1 man or at least 1 woman is 88.6%.