確率

When all outcomes are equally likely, the probability of event $A$ is the number of favourable outcomes divided by the total number of outcomes. For example, the probability of rolling an even number on a fair die is $\frac{3}{6}=\frac{1}{2}$, because $\{2,4,6\}$ gives $3$ favourable outcomes out of $6$. Every probability lies on the probability scale from $0$ (impossible) to $1$ (certain), and $\frac{1}{2}$ represents an even chance.

Since an event either happens or does not, the probability that it happens plus the probability that it does not must add up to $1$. So the probability that $A$ does NOT happen is $P(\text{not }A)=1-P(A)$. For instance, if the probability of rain is $0.3$, the probability of no rain is $1-0.3=0.7$. The complement is often the quickest way to find an awkward probability.

Two events are mutually exclusive if they cannot both happen at the same time. For example, on a single die, 'rolling a $2$' and 'rolling a $5$' cannot both occur. For mutually exclusive events, the probability that $A$ OR $B$ happens is found by adding: $P(A\text{ or }B)=P(A)+P(B)$. Because there is no overlap, you simply add the probabilities.

Two events are independent if the outcome of one does not affect the probability of the other. For example, when you toss a coin and then roll a die, the coin result does not change the die probabilities. For independent events, the probability that $A$ AND $B$ both happen is found by multiplying: $P(A\text{ and }B)=P(A)\times P(B)$. Remember: 'or' adds, 'and' multiplies.

A tree diagram represents a two-stage (or multi-stage) experiment using branches. Write the probability of each step on its branch; following the branches to the right gives one outcome. Multiply along a path (you pass through successive independent stages) and add between paths (the different paths are mutually exclusive outcomes). On any single set of branches, the probabilities always sum to $1$.