What Is a Sample Space?

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The collection all of possible outcomes of a probability experiment forms a set that is known as the sample space.

Probability concerns itself with random phenomena or probability experiments. These experiments are all different in nature, and can concern things as diverse as rolling dice or flipping coins. The common thread that runs throughout these probability experiments is that there are observable outcomes.

The outcome occurs randomly, and is unknown prior to conducting our experiment. 

In this set theory formulation of probability the sample space for a problem corresponds to an important set. Since the sample space contains every outcome that is possible, it forms a setting of everything that we can consider. So the sample space becomes the universal set in use for a particular probability experiment.

Common Sample Spaces

Sample spaces abound and are infinite in number. But there are a few that are frequently used for examples in an introductory statistics or probability course. Below are the experiments and their corresponding sample spaces:

  • For the experiment of flipping a coin, the sample space is {Heads, Tails}.  There are two elements in this sample space.
  • For the experiment of flipping two coins, the sample space is {(Heads, Heads), (Heads, Tails), (Tails, Heads), (Tails, Tails) }.  This sample space has four elements.
  • For the experiment of flipping three coins, the sample space is {(Heads, Heads, Heads), (Heads, Heads, Tails), (Heads, Tails, Heads), (Heads, Tails, Tails), (Tails, Heads, Heads), (Tails, Heads, Tails), (Tails, Tails, Heads), (Tails, Tails, Tails) }.  This sample space has eight elements.
  • For the experiment of flipping n coins, where n is a positive whole number, the sample space consists of 2n elements. There are a total of C (n, k) ways to obtain k heads and n - k tails for each number k from 0 to n.
  • For the experiment consisting of rolling a single six-sided die, the sample space is {1, 2, 3, 4, 5, 6}
  • For the experiment of rolling two six-sided dice, the sample space consists of the set of the 36 possible pairings of the numbers 1, 2, 3, 4, 5 and 6.
  • For the experiment of rolling three six-sided dice, the sample space consists of the set of the 216 possible triples of the numbers 1, 2, 3, 4, 5 and 6.
  • For the experiment of rolling n six-sided dice, where n is a positive whole number, the sample space consists of 6n elements.
  • For an experiment of drawing from a standard deck of cards, the sample space is the set that lists all 52 cards in a deck. For this example the sample space could only consider certain features of the cards, such as rank or suit.

Forming Other Sample Spaces

The above list includes some of the most commonly used sample spaces. Others are out there for different experiments. It is also possible to combine several of the above experiments. When this is done, we end up with a sample space that is the Cartesian product of our individual sample spaces. We can also use a tree diagram to form these sample spaces.

For example, we may want to analyze a probability experiment in which we first flip a coin and then roll a die.

  Since there are two outcomes for flipping a coin and six outcomes for rolling a die, there are a total of 2 x 6 = 12 outcomes in the sample space we are considering.