Combined events and sample space
A combined event is formed by two or more single events happening together, such as tossing a coin and rolling a die. The sample space $S$ is the set of all possible outcomes. If the first event has $m$ outcomes and the second has $n$ outcomes, the combined sample space has $m \times n$ outcomes.
Key formula
$n(S) = m \times n, \qquad P(\text{event}) = \frac{n(\text{favourable})}{n(S)}$
Listing outcomes
You can list outcomes systematically with an ordered list, a two-way table, or a tree diagram.
Example
Toss two coins. The sample space is $\{HH, HT, TH, TT\}$, so $n(S)=4$. The event “exactly one head” is $\{HT, TH\}$, so $P = \frac{2}{4} = \frac{1}{2}$.
Tree diagrams
A tree diagram shows each stage as a set of branches. Tossing a coin twice gives $2 \times 2 = 4$ end branches, one for each outcome. Reading along a path from left to right gives one complete outcome of the combined event.
Example
Roll two dice. Each die has $6$ outcomes, so $n(S) = 6 \times 6 = 36$. The event “sum $= 7$” has outcomes $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$, so $P = \frac{6}{36} = \frac{1}{6}$.
Remember
Always count the total outcomes $n(S)$ first, then count only the favourable ones. Multiply the number of outcomes of each stage to get $n(S)$.