Background/Context: I tried to solve the question
For a certain probability experiment, the probability that event $A$ will occur is $\frac 12$ and the probability that event $B$ will occur is $\frac 13$. Which of the following values could be the probability that the event $A \bigcup B$ will occur? (select all that apply) $a) \frac 13$ $b) \frac 12$ $c) \frac 34$
Since the question gives no information about the dependency I assume both extreme cases:
[Independent] $p(A \bigcup B) = p(A) + p(B) = \frac 12 + \frac 13 = \frac 56$
[Dependent] $p(A \bigcup B) = p(A) + p(B)-p(A \bigcap B) = \frac 56 - p(A \bigcap B)$
Here is where I went wrong, assuming $p(A \bigcap B)$ can take on any value between $0$ and $\frac 56$, making all the choices possible. An explanation states: "The lower limit will occur when the events are dependent and therefore p(A and B) will equal the smallest of the two probabilities. Thus, the lower limit will be whichever probability is greater between p(A) and p(B)"
Midway through the question I thought $p(A \bigcap B)$ can be any value because I just imagined the intersection between two sets on a Venn diagram getting bigger till they completely overlap. I now realise that's completely wrong since a Venn diagram illustrates whether the events are exclusive or not.
But I can't visualize or come up with an example that demonstrates the explanation. How can I see that when events are completely dependent the result will be the larger probability of the two events? Can someone explain?