Random experiments and sample spaces

A random experiment (or just experiment) is a procedure with the following properties:

1. It can be repeated under essentially identical conditions.
2. The outcome on any single trial cannot be predicted with certainty.
3. All possible outcomes are known in advance.

Tossing a coin, rolling a die, drawing a card , all classic examples. The act of asking "who will win the football match?" is not a random experiment in the technical sense (it's not repeatable under identical conditions); but tossing a balanced coin to start the match is.

Sample space

The sample space $S$ of a random experiment is the set of all possible outcomes. We always start by writing $S$ explicitly.

Some standard examples:

• Coin toss. $S = \{H, T\}$.
• Two coin tosses. $S = \{HH, HT, TH, TT\}$.
• Roll a die. $S = \{1, 2, 3, 4, 5, 6\}$.
• Roll two dice. $S = \{(i, j) : 1 \le i, j \le 6\}$, $|S| = 36$.
• Draw a card from a deck. $|S| = 52$.
• Toss a coin until heads. $S = \{H, TH, TTH, TTTH, \dots\}$ , infinite but countable.

Outcomes vs events

A sample point is a single element of $S$ , one outcome.

An event is a subset of $S$. The event "even number on a die" is $\{2, 4, 6\}$. The event "at least one head in two tosses" is $\{HH, HT, TH\}$.

Equally likely outcomes

We often assume each outcome in $S$ is equally likely (a fair coin, a balanced die, a well-shuffled deck). This assumption is crucial for the classical probability formula.

If outcomes are not equally likely (a loaded die, a biased coin), we must assign probabilities to individual outcomes (next subtopic).

Worked examples

Example 1. Write the sample space for tossing three coins.

$S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$. $|S| = 8$.

Example 2. Write the sample space for the experiment: roll a die and toss a coin.

$S = \{(1, H), (1, T), (2, H), (2, T), \dots, (6, T)\}$. $|S| = 12$.

Example 3. A bag contains $3$ red and $2$ blue balls. Draw two without replacement. Write the sample space.

Label balls $R_1, R_2, R_3, B_1, B_2$. $S$ = all unordered pairs: $\{\{R_1, R_2\}, \{R_1, R_3\}, \{R_1, B_1\}, \dots\}$.

$|S| = \binom{5}{2} = 10$.

If order matters (drawing one at a time), $|S| = 5 \cdot 4 = 20$ ordered pairs.

Example 4. Roll two dice. Describe the events: (a) sum = $7$. (b) doubles. (c) sum > $10$.

(a) $\{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\}$. $6$ outcomes. (b) $\{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\}$. $6$ outcomes. (c) $\{(5,6), (6,5), (6,6)\}$. $3$ outcomes.

Example 5. A coin is tossed repeatedly until either heads appears or it has been tossed $5$ times. Sample space?

$S = \{H, TH, TTH, TTTH, TTTTH, TTTTT\}$. $|S| = 6$.

Try it yourself

1. Sample space for two coin tosses.
2. Sample space for rolling a die and noting whether the result is even or odd.
3. Sample space for drawing two cards from a deck (with replacement vs without).
4. Sample space when a fair coin is tossed and then a die rolled.
5. Three students sit for an exam; each either passes (P) or fails (F). Write the sample space.
6. Sample space for rolling three dice and noting the sum.
7. List the event "exactly one head" in three coin tosses.
8. List the event "both even" when two dice are rolled.
9. A card is drawn from a deck and its suit recorded. $|S|$?
10. A box has $5$ white and $3$ black balls. Two balls are drawn together. $|S|$?
11. Toss four coins. List the event "exactly two heads".
12. A die is rolled $n$ times. $|S| = ?$.

Pitfalls / Tricks

• For ordered outcomes use tuples $(i, j)$; for unordered use sets $\{i, j\}$.
• "Equally likely" is a modelling assumption , verify it applies.
• For sequential experiments, sample points are sequences.
• Insight. A well-defined sample space is the most important step. If $S$ is wrong, every subsequent calculation is wrong.