I will use an example to showcase my confusion:
Suppose a person watches show A 2/3 of the time, show B 1/2 of the time, and both show A and show B 1/3 of the time. For a randomly selected day, what is the probability that the person watches only show A? For a randomly selected day, what is the probability that the person watches neither show?
A probability measure [itex]P[/itex] is such that if [itex]A_1, A_2, \cdots[/itex] is a finite or countable sequence of pairwise disjoint events, then [itex]P(A_1 \cup A_2 \cup \cdots) = P(A_1) + P(A_2) + \cdots[/itex]
The Attempt at a Solution
Maybe I'm looking too deeply, or I'm just confused, but here it is...
My first concern is that, if I define [itex]A[/itex] to be the event "the person watches show A," and [itex]B[/itex] the event "the person watches show B," then I have [itex]P(A) + P(B) > 1[/itex]. (These numbers are what was provided.) Does this mean that they cannot be part of the same sample space? Is that okay?
My second concern is that, to my understanding, the required event is [itex]A \cap B^C[/itex], where [itex]B^C[/itex] is the complement of [itex]B[/itex]. To find this, we can write [itex]A = (A \cap B^C) \cup (A \cap B)[/itex] and then apply the additivity property due to how they are disjoint. But, contrary to how the problem has [itex]P(A \cap B) = 1/3[/itex], in terms of set theory, wouldn't this be [STRIKE]∅[/STRIKE] [itex]P(∅)=0[/itex], due to how [itex]A[/itex] and [itex]B[/itex] are disjoint? In my textbook, there was never a formal definition of what the intersection of "events" is, but in my class our professor defined it, maybe informally, to be "the event that both events occur." What definition should I be using?
Anyway, if I go along and solve it ignoring these aspects, you get 1/3 as the probability measure of the first required event, i.e. of [itex]A \cap B^C[/itex].
Moving onto the second question, I become confused. Would this event (the event the person watches neither shows) be [itex]A^C \cap B^C[/itex], leaving room for other possible events not discussed, or just ∅, if I make the sample space only consisting of these events, even though that brings me back to my first problem?
Thanks in advance.