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## Homework Statement

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?

## Homework Equations

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.