Respuesta :
Answer:
0.3846 = 38.46% probability that the subcommittee consists of 2 men and 1 woman.
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
To solve this question, we need to know the combinations formula and conditional probability.
Combinations formula:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
Conditional Probability
We use the conditional probability formula to solve this question. It is
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]
In which
P(B|A) is the probability of event B happening, given that A happened.
[tex]P(A \cap B)[/tex] is the probability of both A and B happening.
P(A) is the probability of A happening.
What is the probability that the subcommittee consists of 2 men and 1 woman, given that it contains both men and women?
Event A: Contains both men and woman
Event B: Consists of two men and 1 woman.
Probability of containing both men and woman:
Desired outcomes:
2 men(from a set of 6) and 1 woman(from a set of 9), or 1 men(from a set of 6) and 2 women(from a set of 9). So
[tex]D = C_{6,2}*C_{9,1} + C_{6,1}*C_{9,2} = \frac{6!}{2!4!}*\frac{9!}{1!8!} + \frac{6!}{1!5!}*\frac{9!}{2!7!} = 351[/tex]
Total outcomes:
3 people from a set of 6 + 9 = 15. So
[tex]T = \frac{15!}{3!12!} = 455[/tex]
Probability:
[tex]P(A) = \frac{D}{T} = \frac{351}{455} = 0.7714[/tex]
Intersection of events A and B:
Intersection between both men and women, and 2 men and 1 woman, is 2 man and 1 women. So
Desired outcomes:
2 men(from a set of 6) and 1 woman(from a set of 9).
[tex]D = C_{6,2}*C_{9,1} = 135[/tex]
Total outcomes:
3 people from a set of 6 + 9 = 15. So
[tex]T = \frac{15!}{3!12!} = 455[/tex]
Probability:
[tex]P(A \cap B) = \frac{D}{T} = \frac{135}{455} = 0.2967[/tex]
Desired probability:
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.2967}{0.7714} = 0.3846[/tex]
0.3846 = 38.46% probability that the subcommittee consists of 2 men and 1 woman.
