Respuesta :
We take the option of writing the answers as fractions.
Answer:
(a) [tex] \\ P(M) = \frac{74.6}{139.4}[/tex]; (b) [tex] \\ P(MJ) = \frac{49.9}{139.4}[/tex]; (c) [tex] \\ P(M \cap MJ) = \frac{24.5}{139.4}[/tex]; (d) [tex] \\ P(MJ|M) = \frac{24.5}{74.6}[/tex]; (e) [tex] \\ P(M|MJ) = \frac{24.5}{49.9}[/tex].
Step-by-step explanation:
This is a case of conditional probability. The general formula for this is as follows:
[tex] \\ P(A|B) = \frac{P(A \cap B)}{P(B)}[/tex]
Looking to the formula, it is important to see that P(B) replaces the total sample space; in other words, and roughly speaking, it is the probability that the event A occurs given (or assuming) that event B happens.
In this kind of problems, it helps doing a kind of contingency table to represent the all the probabilities related to the problem, as follows (in millions):
Management Job Employed
Women 25.4 64.8
Men 24.5 74.6
Total 49.9 139.4
As you can see, the final value is the sum of the previous elements.
It is important to remember this "An employed person is chosen at random."
Why? The employed people is the sample space or the total possible cases here.
We are not going to use units (millions). Probabilities use no units.
Having prepared all this information, we have the elements to solve the questions.
(a) What is the probability that the person is a male?
From the table, from the employed people, 74.6 are men.
So
[tex] \\ P(M) = \frac{74.6}{139.4}[/tex]
b) What is the probability that the person has a management job?
The total of men and women with a Management Job is 49.9.
So
[tex] \\ P(MJ) = \frac{49.9}{139.4}[/tex]
(c) What is the probability that the person is male and has a management job?
Here, we are dealing with the probability that the person is Male AND has a Management Job. The number of possible cases of the event that is a men and has a management job is 24.5.
So
[tex] \\ P(M \cap MJ) = \frac{24.5}{139.4}[/tex]
Using conditional probability formula, we can obtain the same result.
[tex] \\ P(M|MJ) = \frac{P(M \cap MJ)}{P(MJ)}[/tex]
[tex] \\ P(M \cap MJ) = P(M|MJ) * P(MJ)[/tex]
[tex] \\ P(M \cap MJ) = \frac{24.5}{49.9}*\frac{49.9}{139.4}[/tex]
[tex] \\ P(M \cap MJ) = \frac{24.5}{139.4}[/tex]
(d) Given that the person is male, what is the probability that he has a management job?
It is a conditional probability as follows:
[tex] \\ P(MJ|M) = \frac{P(M \cap MJ)}{P(M)}[/tex]
The probability P(M) assumes the sample space in this case. Considering the values of the table, we have the following result.
The event Men AND having a Management Job equals 24.5. The total men employed equals 74.6. Thus, we can write it as:
[tex] \\ P(MJ|M) = \frac{P(M \cap MJ)}{P(M)}[/tex]
[tex] \\ P(MJ|M) = \frac{\frac{24.5}{139.4}}{\frac{74.6}{139.4}}[/tex]
[tex] \\ P(MJ|M) = \frac{24.5}{74.6}[/tex]
(e) Given that the person has a management job, what is the probability that the person is male?
The conditional probability has a similar "structure" than the previous question, and it is as follows:
[tex] \\ P(M|MJ) = \frac{P(M \cap MJ)}{P(MJ)}[/tex]
The event Men AND having a Management Job equals 24.5. The total for management job is 49.9. Therefore:
[tex] \\ P(M|MJ) = \frac{\frac{24.5}{139.4}}{\frac{49.9}{139.4}}[/tex]
[tex] \\ P(M|MJ) = \frac{24.5}{49.9}[/tex]
