What it is
A random experiment is a procedure whose result is decided by chance but whose possible results can be listed in advance. Rolling a die is a random experiment. The result of one trial is an outcome, for example “four”, and all the possible outcomes together are the sample space, often written Ω. For a die, Ω = {1, 2, 3, 4, 5, 6}.
An event is a collection of outcomes, that is, a subset of the sample space. “Even number” is the event {2, 4, 6}, and it happens if the outcome is one of those three numbers. A single outcome is also an event, {4}. The empty event can never happen, and the whole of Ω always does. Events are sets and are combined like sets.
| Random experiment | Sample space (H = heads, T = tails) | Number of outcomes |
|---|---|---|
| One coin toss | {H, T} | 2 |
| One roll of a die | {1, 2, 3, 4, 5, 6} | 6 |
| Two coin tosses | {HH, HT, TH, TT} | 4 |
| Three coin tosses | HHH, HHT, HTH, HTT, THH, THT, TTH, TTT | 8 |
| Two rolls of a die | all pairs (1, 1), (1, 2), …, (6, 6) | 36 |
| One card from a deck | the 52 cards | 52 |
The number of outcomes is counted with the multiplication principle: three tosses give 2 · 2 · 2 = 8 outcomes and two dice 6 · 6 = 36.
The same experiment can be described with different sample spaces, depending on what you record. A roll of a die can have the sample space {1, 2, 3, 4, 5, 6} or {six, not six}. Both will do, but only in the first are the outcomes equally likely.
A sample space need not be a short list. In the experiment “roll until a six appears” the outcomes are the number of rolls, 1, 2, 3 and so on without end, and when waiting for a bus that runs once an hour the outcomes are all the times between 0 and 60 minutes.
Not experiments and not outcomes: 2 + 3 is not a random experiment, because the answer is fixed in advance. “Even number” and “at least 4” are not outcomes but events.
Why it matters
The sample space is the denominator of classical probability. When all outcomes are equally likely,
P(A) = number of outcomes in A / number of outcomes in Ω
as on the probability of an event page. With the wrong sample space, the answer is wrong.
- The sum of two dice. The sums 2, 3, …, 12 are eleven possible results, but they are not equally likely. Treating the sums as outcomes gives P(sum 7) = 1/11 ≈ 0.091, which is wrong. The 36 pairs are equally likely, and six of them give the sum 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), so P(sum 7) = 6/36 = 1/6 ≈ 0.167.
- Two coin tosses. The probability of exactly one head is 2/4 = 1/2, because HT and TH are different outcomes. Treating 0, 1 and 2 heads as three equally likely outcomes gives 1/3; their probabilities are 1/4, 1/2 and 1/4.
- Complement, union and intersection. A not happening is the event Ω without A; see the complement rule. “A or B” is the union and “A and B” the intersection; see the addition rule.
- Counting outcomes. Ordered choices are counted with permutations and unordered ones with combinations. A lottery that draws 7 numbers out of 35 has C(35, 7) = 6,724,520 outcomes, so the probability of all seven on one line is 1/6,724,520.
- Measuring instead of counting. When the outcomes are not equally likely, as with an uneven die, the probabilities are estimated by repeating the experiment and counting the relative frequency.
- The total is 1. In a sample space with finitely many outcomes, the probabilities of the outcomes add up to 1. Two dice have 36 outcomes with probability 1/36 each.
Common misconceptions
- “Two outcomes means 50/50.” “Six” and “not six” are two outcomes with probabilities 1/6 and 5/6. Favourable over possible holds only when the outcomes are equally likely. Win, draw and loss are three outcomes of a match, but nothing says they are equally likely.
- “HT and TH are the same outcome.” In the sample space for two tosses they are different, because the first and the second toss are different tosses. Merged into “one head and one tail” they make an event with two outcomes.
- “Outcome and event mean the same thing.” An outcome is one result; an event is a collection of results. “Even number” is the event {2, 4, 6} and consists of three outcomes.
- “The sample space is the outcomes I want.” The sample space is all the possible outcomes. The favourable ones make up the event, a subset of the sample space.
Common questions
How many outcomes do three coin tosses have, and how many of them give exactly two heads?
Eight outcomes, 2 · 2 · 2. Three of them have exactly two heads: HHT, HTH and THH. The probability is
3/8 = 0.375.
Can the sample space be infinite?
Yes. The number of rolls until the first six can be 1, 2, 3 and so on without limit, and a waiting time can be
any number in an interval. Then favourable over possible no longer works, and the probabilities come from a
distribution instead, for example the
geometric distribution for the number of rolls
until the first six.
What does Ω mean?
The Greek letter omega, often used for the sample space. Some books write S instead.