3. Probability: Randomness
Sample Space
In general, it is not possible to predict the outcome of an experiment with certainty. It is possible, however, to make a list of all potential outcomes. Such a list is called the sample space.
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Sample space
Definition
The sample space of an experiment is the set with all possible outcomes of that experiment as its elements.
The sample space is denoted by #\Omega#.
Examples
- Tossing a coin
- #\Omega = \{#Heads, Tails#\}#
- Rolling a die
- #\Omega = \{#1, 2, 3, 4, 5, 6#\}#
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It is common practice to abbreviate the outcomes of an experiment.
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The experiment of tossing a coin one time has 2 possible outcomes:
- H = coin comes up Heads
- T = coin comes up Tails
So the sample space of this experiment is #\Omega = \{#H, T#\}#.
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It is also possible to form a new random experiment by repeating an experiment. An example would be tossing a coin twice instead of just once.
The sample space of a repeated experiment is the set of all the possible different combinations of outcomes of the original experiment.
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The experiment of tossing a coin twice has 4 possible outcomes:
- (H, T): first heads, then tails
- (T, H): first tails, then heads
- (H, H): two times heads
- (T, T): two times tails
So the sample space of this experiment is #\Omega = \{#HT, TH, HH, TT#\}#.
Note that the outcomes are written in a specific order:
- (H, T) indicates Heads on the first coin toss and Tails on the second
- (T, H) indicates you get Tails on the first coin toss and Heads on the second
These two outcomes are different from each other and should, therefore, both be considered as unique outcomes of the repeated experiment.
