I think I am missing something very basic.
What does it mean when people say probability of something "on an event"? Is it different from conditional probability?
For example, suppose I have an event $D$, and I know that if event $E$ does not occur, then $D$ must occur. Is this the same as saying that $P(D) = 1$ on $\{E^c\}$, if so, how is it different from $P(D\mid E^c) = 1$?
Edit: I understand that “given an event” = conditional probability. I am trying to understand if probability “on an event” is the same concept, or if it has s different definition.
$\Bbb P(D \mid E^c)$ is the conditional probability of $D$ given $E^c.$ If that event is certain that means $D \cap E^c = E^c.$ Therefore $$\Bbb P(D \mid E^c) = \frac {\Bbb P(D \cap E^c)} {\Bbb P(E^c)} = \frac {\Bbb P(E^c)} {\Bbb P(E^c)} =1.$$
Note that $D \mid E^c$ denotes the event of occurrence of $D$ when $E^c$ has already occurred. So in that case the sample space reduces to $E^c.$ If $D \mid E^c$ is certain that means $E^c \subseteq D.$