Calculate Outcome of the Probability of the Union of Two Events

Question

I am working on gathering statistics about some values I observe. I will try to explain as best as I can

I have gathered enough information and what is the supposed to be the outcome when one event occurs. I have two events and two outcomes. Say that I am having EventA,EventB and OutcomeA,OutcomeB. What is the most probable Outcome when the two Events come together. Is there a way to find calculate the direction of the outcome?

I am aware with the basics on how to Compute the Probability of the Union of Two Events. But I cannot seem to figure it out.

The problem to solve

I am having two events. EventA has 70% probability for it to have the OutcomeA and 30% for OutcomeB. On the other hand EventB has 60% for OutcomeA and 40% for OutcomeB.

What is the probability for OutcomeA and/or OutcomeB when EventA and EventB occur at the same time?

Answer 1

You can see it as you have two events $A, B$, and their probabilities are

$P(A) = 0.7$ $\Rightarrow$ $P(A^c) = 0.3$ and $P(B) = 0.4$ $\Rightarrow$ $P(B^c) = 0.6$.

And you want the probability of them happening at the same time to obtain the probabilities of both outcomes:

$P(A\cap B)$ and $P(A^c \cap B^c)$, wich you can get with the definition of conditional probability.