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:
![\\ P(A|B) = \frac{P(A \cap B)}{P(B)}](https://tex.z-dn.net/?f=%20%5C%5C%20P%28A%7CB%29%20%3D%20%5Cfrac%7BP%28A%20%5Ccap%20B%29%7D%7BP%28B%29%7D)
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:
![\\ P(T|M) = 0.47](https://tex.z-dn.net/?f=%20%5C%5C%20P%28T%7CM%29%20%3D%200.47)
The probability that a person has a tattoo assuming that is of Generation X is:
![\\ P(T|X) = 0.36](https://tex.z-dn.net/?f=%20%5C%5C%20P%28T%7CX%29%20%3D%200.36)
The probability that a person has a tattoo assuming that is of Boomers is:
![\\ P(T|B) = 0.13](https://tex.z-dn.net/?f=%20%5C%5C%20P%28T%7CB%29%20%3D%200.13)
The probability of being of Millennials is:
![\\ P(M) = 0.22](https://tex.z-dn.net/?f=%20%5C%5C%20P%28M%29%20%3D%200.22)
The probability of being of Generation X is:
![\\ P(X) = 0.20](https://tex.z-dn.net/?f=%20%5C%5C%20P%28X%29%20%3D%200.20)
The probability of being of Boomers is:
![\\ P(B) = 0.22](https://tex.z-dn.net/?f=%20%5C%5C%20P%28B%29%20%3D%200.22)
Therefore, the probability of the event of having a tattoo P(T) is:
![\\ P(T) = P(T|M)*P(M) + P(T|X)*P(X) + P(T|B)*P(B)](https://tex.z-dn.net/?f=%20%5C%5C%20P%28T%29%20%3D%20P%28T%7CM%29%2AP%28M%29%20%2B%20P%28T%7CX%29%2AP%28X%29%20%2B%20P%28T%7CB%29%2AP%28B%29)
![\\ P(T) = 0.47*0.22 + 0.36*0.20 + 0.13*0.22](https://tex.z-dn.net/?f=%20%5C%5C%20P%28T%29%20%3D%200.47%2A0.22%20%2B%200.36%2A0.20%20%2B%200.13%2A0.22)
![\\ P(T) = 0.204](https://tex.z-dn.net/?f=%20%5C%5C%20P%28T%29%20%3D%200.204)
For non-independent events that happen at the same time, we can say that the probability of occurring simultaneously is:
![\\ P(M \cap T) = P(M|T)*P(T)](https://tex.z-dn.net/?f=%20%5C%5C%20P%28M%20%5Ccap%20T%29%20%3D%20P%28M%7CT%29%2AP%28T%29)
Or
![\\ P(T \cap M) = P(T|M)*P(M)](https://tex.z-dn.net/?f=%20%5C%5C%20P%28T%20%5Ccap%20M%29%20%3D%20P%28T%7CM%29%2AP%28M%29)
But
![\\ P(M \cap T) = P(T \cap M)](https://tex.z-dn.net/?f=%20%5C%5C%20P%28M%20%5Ccap%20T%29%20%3D%20P%28T%20%5Ccap%20M%29)
Then
![\\ P(M|T)*P(T) = P(T|M)*P(M)](https://tex.z-dn.net/?f=%20%5C%5C%20P%28M%7CT%29%2AP%28T%29%20%3D%20P%28T%7CM%29%2AP%28M%29)
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:
![\\ P(M|T)*P(T) = P(T|M)*P(M)](https://tex.z-dn.net/?f=%20%5C%5C%20P%28M%7CT%29%2AP%28T%29%20%3D%20P%28T%7CM%29%2AP%28M%29)
![\\ P(M|T) = \frac{P(T|M)*P(M)}{P(T)}](https://tex.z-dn.net/?f=%20%5C%5C%20P%28M%7CT%29%20%3D%20%5Cfrac%7BP%28T%7CM%29%2AP%28M%29%7D%7BP%28T%29%7D)
We have already know that
.
Therefore
![\\ P(M|T) = \frac{0.47*0.22}{0.204}](https://tex.z-dn.net/?f=%20%5C%5C%20P%28M%7CT%29%20%3D%20%5Cfrac%7B0.47%2A0.22%7D%7B0.204%7D)
![\\ P(M|T) = 0.50686 \approx 0.51](https://tex.z-dn.net/?f=%20%5C%5C%20P%28M%7CT%29%20%3D%200.50686%20%5Capprox%200.51)
Thus, the probability that a person is a Millennial given that they have tattoos is 0.5069 (50.69%) or about 0.51 (51%).