### Chapter 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
*H**eads* - T = coin comes up
*T**ails*

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.

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