Event

An event is a subset of the sample space, named with a capital letter such as $A$ or $B$. An event holding a single outcome is called simple, one holding several is compound. Add up the probabilities of every outcome it contains and you have $P(A)$, so when those outcomes are equally likely the adding collapses into counting.

Roll one fair die. Let $A$ be "the roll is even", the set $\{2, 4, 6\}$, and let $B$ be "the roll is greater than 3", the set $\{4, 5, 6\}$. Then $P(A) = 3/6 = 0.5$ and $P(B) = 3/6 = 0.5$. Events combine into new events: "A and B" is $\{4, 6\}$ with probability $2/6 \approx 0.3333$, and "A or B" is $\{2, 4, 5, 6\}$ with probability $4/6 \approx 0.6667$.

"A or B means one of them but not both." Read that way, "A or B" would be $\{2, 5\}$ with probability $2/6 \approx 0.3333$, which is wrong. In probability "or" is inclusive: the event holds whenever at least one of $A$ and $B$ holds, so 4 and 6 stay in. The mirror-image slip is to add, $0.5 + 0.5 = 1$, which claims every roll satisfies "A or B" when a 1 or a 3 satisfies neither. The addition rule exists to strip out the outcomes counted twice.

Not every event can be handled by listing. "The mean of 40 measurements exceeds 12" is an event on a sample space far too large to write down, and its probability comes from a model rather than a count. At the two extremes, the empty set and the whole sample space are both events, with probabilities 0 and 1.

Events are what the rest of Unit 2 operates on. The complement of an event, the union of two events, and their intersection are all events themselves, which is what lets the rules be chained together.