Respuesta :
Answer:
The probability that a person is a Millennial given that they have tattoos is 0.5069 (50.69%) or about 0.51 (51%).
Step-by-step explanation:
We have here a case where we need to use Bayes' Theorem and all conditional probabilities related. Roughly speaking, a conditional probability is a kind of probability where an event determines the occurrence of another event. Mathematically:
[tex] \\ P(A|B) = \frac{P(A \cap B)}{P(B)}[/tex]
In the case of the Bayes' Theorem, we have also a conditional probability where one event is the sum of different probabilities.
We have a series of different probabilities that we have to distinguish one from the others:
The probability that a person has a tattoo assuming that is a Millennial is:
[tex] \\ P(T|M) = 0.47[/tex]
The probability that a person has a tattoo assuming that is of Generation X is:
[tex] \\ P(T|X) = 0.36[/tex]
The probability that a person has a tattoo assuming that is of Boomers is:
[tex] \\ P(T|B) = 0.13[/tex]
The probability of being of Millennials is:
[tex] \\ P(M) = 0.22[/tex]
The probability of being of Generation X is:
[tex] \\ P(X) = 0.20[/tex]
The probability of being of Boomers is:
[tex] \\ P(B) = 0.22[/tex]
Therefore, the probability of the event of having a tattoo P(T) is:
[tex] \\ P(T) = P(T|M)*P(M) + P(T|X)*P(X) + P(T|B)*P(B)[/tex]
[tex] \\ P(T) = 0.47*0.22 + 0.36*0.20 + 0.13*0.22[/tex]
[tex] \\ P(T) = 0.204[/tex]
For non-independent events that happen at the same time, we can say that the probability of occurring simultaneously is:
[tex] \\ P(M \cap T) = P(M|T)*P(T)[/tex]
Or
[tex] \\ P(T \cap M) = P(T|M)*P(M)[/tex]
But
[tex] \\ P(M \cap T) = P(T \cap M)[/tex]
Then
[tex] \\ P(M|T)*P(T) = P(T|M)*P(M)[/tex]
We are asked for the probability that a person is a Millennial given or assuming that they have tattoos or P(M | T). Solving the previous formula for the latter:
[tex] \\ P(M|T)*P(T) = P(T|M)*P(M)[/tex]
[tex] \\ P(M|T) = \frac{P(T|M)*P(M)}{P(T)}[/tex]
We have already know that
[tex] \\ P(T|M) = 0.47\;P(M) = 0.22\;and\;P(T) = 0.204[/tex].
Therefore
[tex] \\ P(M|T) = \frac{0.47*0.22}{0.204}[/tex]
[tex] \\ P(M|T) = 0.50686 \approx 0.51[/tex]
Thus, the probability that a person is a Millennial given that they have tattoos is 0.5069 (50.69%) or about 0.51 (51%).
