Most probability problems appear in middle school and high school competitions are countable, i.e. the number of events can be counted. As such, these probability problems are closely related to counting:

1. Count the total possible events
2. Count the qualified events (i.e. those meet the requirements)
3. Compute the ratio of (2) to (1) which is the desired probability

Let $\mathcal{A}$ be an event, then we use $P(\mathcal{A})$ to indicate the probability that the event $\mathcal{A}$ occurs.

• For any event $\mathcal{A}$, it always holds that $0\le P(\mathcal{A}) \le 1$.

Two important concepts in probability are independent events and mutually exclusive events.

• Two events are independent means that whether or not one event occurs has no correlation with the occurrence of the other event.
• Two events are mutually exclusive means that they cannot occur simultaneously

These lead to two important conclusions:

• If $\mathcal{A}$ and $\mathcal{B}$ are independent, then the probability they both occur is $P(\mathcal{A})\cdot P(\mathcal{B})$
• If $\mathcal{A}$ and $\mathcal{B}$ are mutually exclusive, then the probability that at least one of them occurs is $P(\mathcal{A}) +P(\mathcal{B})$

Please note the similarity between probability calculation and the inclusion and exclusion principle. Let $P(\mathcal{A}\cup\mathcal{B})$ be the probability that at least one of these two events occurs, and $P(\mathcal{A}\cap\mathcal{B})$ be the probability that both of them occur. Then we have $$P(\mathcal{A}\cup\mathcal{B})=P(\mathcal{A}) + P(\mathcal{B}) - P(\mathcal{A}\cap\mathcal{B})$$

Additionally, for events $\mathcal{A}$ and $\mathcal{B}$:

• They are independent is equivalent to saying $P(\mathcal{A}\cap\mathcal{B})=P(\mathcal{A})\cdot P(\mathcal{B})$
• They are mutually exclusive is equivalent to saying $P(\mathcal{A}\cap\mathcal{B})=0$

Tip: many counting techniques (such as count the opposite etc) can also be used to calculate probability.