Explain why the exponents cannot be added in the product 12^3 and<br><br> 11^3

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:

$\\ P(A|B) = \frac{P(A \cap B)}{P(B)}$

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:

$\\ P(T|M) = 0.47$

The probability that a person has a tattoo assuming that is of Generation X is:

$\\ P(T|X) = 0.36$

The probability that a person has a tattoo assuming that is of Boomers is:

$\\ P(T|B) = 0.13$

The probability of being of Millennials is:

$\\ P(M) = 0.22$

The probability of being of Generation X is:

$\\ P(X) = 0.20$

The probability of being of Boomers is:

$\\ P(B) = 0.22$

Therefore, the probability of the event of having a tattoo P(T) is:

$\\ P(T) = P(T|M)*P(M) + P(T|X)*P(X) + P(T|B)*P(B)$

$\\ P(T) = 0.47*0.22 + 0.36*0.20 + 0.13*0.22$

$\\ P(T) = 0.204$

For non-independent events that happen at the same time, we can say that the probability of occurring simultaneously is:

$\\ P(M \cap T) = P(M|T)*P(T)$

Or

$\\ P(T \cap M) = P(T|M)*P(M)$

But

$\\ P(M \cap T) = P(T \cap M)$

Then

$\\ P(M|T)*P(T) = P(T|M)*P(M)$

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:

$\\ P(M|T)*P(T) = P(T|M)*P(M)$

$\\ P(M|T) = \frac{P(T|M)*P(M)}{P(T)}$