Random Experiment and Sample Space
Start with intuition
Think about flipping a coin. Before you flip it, you know the only possible outcomes are heads or tails. But you cannot say which one will actually come up — that depends on chance. That's the core idea: a random experiment is any process where you know what could happen, but you don't know exactly what will happen in a single trial.
Another example: rolling a die. You know the result will be one of 1, 2, 3, 4, 5, or 6. But before you roll, you cannot predict the exact number. That's randomness.
The sample space is simply the set of all possible outcomes of that random experiment. For the coin, the sample space is {H,T}. For the die, it's {1,2,3,4,5,6}.
The precise definition
A random experiment satisfies three conditions:
- It can be repeated any number of times under essentially the same conditions.
- All possible outcomes are known in advance.
- The actual outcome on any given trial cannot be predicted with certainty.
The sample space (denoted by S or Ω) is the set of all possible outcomes of a random experiment. Each individual outcome is called a sample point.
The sample space must be exhaustive — it must contain every possible outcome, and nothing outside it can occur.
Examples to cement the idea
| Random Experiment | Sample Space |
|---|
| Toss a coin | S={H,T} |
| Roll a die | S={1,2,3,4,5,6} |
| Toss two coins | S={HH,HT,TH,TT} |
| Roll two dice | S={(1,1),(1,2),…,(6,6)} — 36 outcomes |
| Draw a card from a deck | S={A♠,K♠,…,2♣} — 52 outcomes |
Notice that for two coins, the order matters: HT and TH are different outcomes because the first coin and second coin are distinct objects.
A common mistake
Students often write the sample space for two coins as {HH,HT,TT}, leaving out TH. That is wrong. The two coins are physically distinct (or the same coin tossed twice), so HT and TH are different outcomes. The sample space must have 4 elements, not 3.
Never collapse distinct outcomes just because they look similar. If the experiment distinguishes between the two coins, the sample space must reflect that.
Why this matters
The sample space is the foundation of probability. Every event (a subset of the sample space) is defined in terms of it. If you get the sample space wrong, every probability you compute from it will be wrong. So always ask: What are all the things that could happen? Then list them systematically.