Question about probability without any numbers. I was asked this question on an exam, but I have no idea if there is enough information to actually answer it.
"Which has a higher probability: Multiple highways in California being closed down or multiple highways in California being closed down following an earthquake."
I thought it was the second one, but my professor said it was the first one.
 A: If the second case is true, then necessarily multiple highways in California are closed, which implies that the first case is also true. That is, anytime that multiple highways are closed due to an earthquake, it is true that multiple highways are closed.
Since every instance of the second case is also an instance of the first case, we know that the first is at least as likely as the second. And since there are cases when multiple highways close down which are not caused by earthquakes, the first is strictly more likely.
If you learn probability with the language of sets, then there is some probability space $X$ and you have two events: $A$ (multiple highways are closed), and $B$ (multiple highways are closed due to an earthquake). Then as $B \subset A$, we have that $p(B) \leq p(A)$.
A: This one has a simple answer. Let's break the possibilities into three cases: 
Case 1: Multiple highways in California are not closed down
Case 2: Multiple highways in California are closed down but not because of an earthquake
Case 3: Multiple highways in California are closed down because of an earthquake
The probability of each of these (we call them $P(C_1)$, $P(C_2)$, 
 and $P(C_3)$ respectively) are both above 0 (positive). 
So the probability multiple highways are closed down is: 
$P(C_2)+P(C_3)$
And the probability multiple highways are closed down because of an earthquake is: 
$P(C_3)$
So which is bigger?
Well: 
$$P(C_2)+P(C_3)\stackrel{?}{>}P(C_3) \Longrightarrow P(C_2)\stackrel{?}{>}0$$
Which we already said is true, so the probability multiple highways are closed down is  greater than the probability multiple highways are closed down because of an earthquake. 
Also, the first is a more general case, so this should be intuitively true. 
