LISTING ALL POSSIBLE OUTCOMES (THE SAMPLE SPACE)


Suppose that we will observe some process or experiment in which the outcome is not known in advance. For example, suppose we plan to roll two dice and we're interested in the sum of the two numbers appearing on the top faces. Before we can talk about probabilities of various sums, say 3 or 7, we have to understand what outcomes are possible in this experiment.

If we roll two dice, each die could show 1, 2, 3, 4, 5, 6. So the sum of the two faces could be any whole number from 2 to 12. We call this set of possible outcomes in the random experiment the sample space . Here the sample space can be written as

Sample space = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Let's consider the set of all possible outcomes for other basic random experiments. Suppose we plan to toss a coin 3 times and the outcome of interest is the number of heads. The sample space in this case is the different numbers of heads you could get if you toss a coin three times. Here you could get 0 heads, 1 heads, 2 heads or 3 heads, so we write the sample space as

Sample space = {0, 1, 2, 3}

Don't forget to include the outcome 0 -- if we toss a coin three times and get all tails, then the number of heads is equal to 0.

The concept of a sample space is also relevant for experiments where the outcomes are non-numerical. Suppose I draw a card from a standard deck of playing cards. If I'm interested in the suit of the card, there are four possible outcomes and the sample space is

Sample space = {Spade, Heart, Diamond, Club}

If I am interested in the suit and the face of the card, then there are many possible outcomes. One can represent the sample space by the following table:

                            FACE OF CARD
     SUIT    2    3    4    5    6    7    8    9   10    J    Q    K    A
    -----------------------------------------------------------------------
    Spade    x    x    x    x    x    x    x    x    x    x    x    x    x
    Heart    x    x    x    x    x    x    x    x    x    x    x    x    x
    Diamond  x    x    x    x    x    x    x    x    x    x    x    x    x
    Club     x    x    x    x    x    x    x    x    x    x    x    x    x

Each "x" in the table corresponds to a particular outcome of the experiment. For example, the first "x" in the third row of the table corresponds to a draw of the 2 of Diamonds. We see from this table that there are 52 possible outcomes.

Once we understand what the collection of all possible outcomes looks like, we can think about assigning probabilities to the different outcomes. But be careful -- incorrect probability assignments can be made because in mistakes in specifying the entire sample space.

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Page Author: Jim Albert (©1996)
albert@bayes.bgsu.edu
Document: http://www-math.bgsu.edu/~albert/m115/probability/sample_space.html
Last Modified: January 21 1998