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The sample space is the set of every elementary outcome that can occur in an experiment. We write it with the symbol (Omega).
For one roll of a standard -sided die, the possible outcomes are and . Together they form the sample space:
There are a total of possible outcomes here.
An event is a set of outcomes satisfying a condition. Its probability is the sum of the probabilities of those outcomes. When all elementary outcomes are equally likely, divide the number of favorable outcomes by the total number of outcomes.
An event is a subset of the sample space containing the outcomes that satisfy a condition of interest.
Example:
For the die experiment (), consider the event "the result is even." The even outcomes are and , so event is:
There are outcomes in this event .
Probability measures how likely an event is to occur.
When all elementary outcomes are equally likely, the probability is the number of favorable outcomes divided by the total number of outcomes:
Using the symbols from our example:
Here:
So, the probability of the event "getting an even number" (event ) is:
The event therefore has probability , or .
A probability distribution is valid when both rules hold for every possible value of the random variable. The two conditions make the listed probabilities add up to a complete description of all outcomes.
The probability of each value () lies between and .
The probabilities of all possible values add up to exactly .
For one fair die, every face has the same probability. Its distribution is:
| Outcome | Probability |
|---|---|
Why are they all ?
The die has equally likely faces, so each face has probability . Their sum is , as required.
Now roll one red die and one white die. Listing the result as an ordered pair produces possible outcomes.
Why ?
The red die has possibilities and the white die has possibilities, giving ordered pairs.
The resulting pairs can be written like this:
Because the dice are fair and independent, each ordered pair has probability .
Distinguish from
Because the dice have different colors, a red with a white differs from a red with a white . The order records which die produced each number.
If the random variable is the sum of the two dice, its smallest value is and its largest value is . Different sums contain different numbers of ordered pairs, so their probabilities differ:
| Sum of Numbers | Possible Pairs | Number of Pairs | Probability |
|---|---|---|---|
| , | |||
| , , | |||
| , , , | |||
| , , , , | |||
| , , , , , | |||
| , , , , | |||
| , , , | |||
| , , | |||
| , | |||
The sum is most likely because ordered pairs produce it, while the sums and each come from only one pair. This difference matters when a game or model uses the total from both dice.
A probability distribution pairs each possible outcome with its probability. Those numbers support several concrete comparisons:
Each use relies on the same structure: a defined set of outcomes and a probability assigned to every outcome.
Three fair coins are tossed. Let be the number of heads. Build the probability distribution of , verify that it is valid, and find .
Encode a head as and a tail as . The three tosses then produce equally likely ordered triples:
Count the outcomes that contain or heads:
| Number of heads | Outcomes | Count | |
|---|---|---|---|
Every probability lies between and , and their sum is
The distribution is therefore valid. At least two heads means or , so
Listing the sample space first makes the distribution auditable: each of the outcomes appears in exactly one row.
Published: . Updated: .
