Probability foundations · Definition

Sample space

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Before assigning probabilities we must know what can happen: the sample space is the starting point of the whole theory.

Definition

A sample space \(\Omega\) is the set of all possible outcomes \(\omega\) of a random experiment.

\[\Omega=\{\omega_1,\omega_2,\omega_3,\dots\}\]

A random experiment is one whose outcome cannot be predicted with certainty, even though the possible outcomes are known in advance. The set of all of them is called \(\Omega\), and each element \(\omega\in\Omega\) is an outcome. Outcomes must be exhaustive (none is missing) and mutually exclusive (exactly one occurs when the experiment is performed).

The sample space depends on what we decide to record. When tossing two coins, recording the sequence gives \(\Omega=\{HH,HT,TH,TT\}\), with 4 elements; recording only the number of heads gives \(\Omega=\{0,1,2\}\). Both descriptions are valid, but we should pick the one that answers our question without losing information.

By size, a sample space can be finite (a die), countably infinite (the number of tosses until the first head: \(1,2,3,\dots\)) or uncountably infinite (a waiting time, \([0,\infty)\)). When experiments are combined, counts multiply: two dice give \(6\cdot 6=36\) ordered pairs.

A common mistake is to confuse the sample space with an event: \(\Omega\) contains all possible outcomes, while an event such as «the roll is even» is only a part of it. The next concept formalizes that idea.

Think about it

Two dice are rolled and we only care whether the sum is even. Is \(\Omega=\{\text{even},\text{odd}\}\) enough?

It may suffice for that question, but it hides information and does not allow other probabilities to be computed; the 36 ordered pairs are a finer and more useful description.

Exercises 10

  1. Two distinguishable coins are tossed and the result of each is recorded. How many elements does the sample space have?
  2. A six-sided die is rolled and the number shown is recorded. What is the sample space?
  3. Two distinguishable dice are rolled and the pair of numbers is recorded. How many outcomes does the sample space have?
  4. A coin is tossed three times and the sequence is recorded. How many elements does the sample space have?
  5. Two dice are rolled and only the sum is recorded. How many elements does the sample space have?
  6. A coin is tossed until heads appears and the number of tosses is recorded. What is the sample space like?
  7. The time, in minutes, until a bus arrives is measured. Which sample space is appropriate?
  8. A die and a coin are tossed and the pair (number, side) is recorded. How many elements does the sample space have?
  9. From an urn with three distinct balls (R, B, G) one is drawn, its color recorded and returned; then another is drawn and recorded. How many ordered outcomes does the sample space have?
  10. Which statement about the sample space is correct?

What comes next

References

  • Joseph K. Blitzstein, Jessica Hwang, Introduction to Probability, 2nd edition, CRC Press, 2019. §1.2
  • Sheldon Ross, A First Course in Probability, 10th edition, Pearson, 2019. §2.2
  • Larry Wasserman, All of Statistics, 1st edition, Springer, 2004. §2.1