Sample space
Definition taught in Random experiments and probability, part of Probability
Sample space
| Type | Definition |
| Taught in | Random experiments and probability |
| Part | Probability |
| Properties | 2 |
| Worked examples | 2 |
| Questions using it | 5 |
The sample space \Omega of a random experiment is the set of all outcomes that experiment can produce.
Properties
Equally likely outcomes
Main article: Equally likely outcomes
If \Omega is finite and every outcome has the same probability, then for every event A
\mathbb{P}(A)=\frac{|A|}{|\Omega|}.
Four ways of counting draws
Main article: Four ways of counting draws
| Order recorded | Replacement | Number of outcomes |
|---|---|---|
| yes | yes | n^k |
| yes | no | n!/(n-k)! |
| no | no | \binom{n}{k} |
| no | yes | \binom{n+k-1}{k} |
Ordered outcomes are written as vectors, so (1,2,3)\ne(3,2,1). Unordered outcomes are written as sets, so \{1,2,3\}=\{3,2,1\}.
Examples
Three coin tosses
Question page: Three coin tosses
A fair coin is tossed three times.
List the sample space if the whole sequence of heads and tails is recorded. Write heads as 1 and tails as 0.
List the sample space if only the number of heads is recorded.
- The sample space is every sequence of length 3 with coordinates in \{0,1\}:
\Omega=\{(0,0,0),(0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(1,1,0),(1,1,1)\}.
- If only the number of heads is kept, \Omega=\{0,1,2,3\}.
Weights on the two coin spaces
Question page: Weights on the two coin spaces
A fair coin is tossed three times, and the tosses are independent. Assign a probability to every outcome of each of these two sample spaces: the set of sequences in \{0,1\}^3, and the set \{0,1,2,3\} of possible numbers of heads.
Independence and fairness give probability 1/8 to each of the eight sequences.
The number of heads is binomial. \mathbb{P}(\{0\})=\mathbb{P}(\{3\})=1/8 and \mathbb{P}(\{1\})=\mathbb{P}(\{2\})=3/8.
Used in
- Taught in Random experiments and probability, where the derivation and examples are.
- Also used in Random variables and distributions
- Questions that use it (5) — Three coin tosses · Weights on the two coin spaces · Three cards in order · One of each court card (answer) · The sum of two dice
- All 17 questions on Random experiments and probability
See also
Event · Random variable · Probability measure · Properties of a probability · Equally likely outcomes · Four ways of counting draws · Conditional probability · Product rule