Sample Space

Randomness or the chance is the basis for the study of probability. A probability experiment gives out well defined random outcomes.

The probability experiment is also called a random experiment.

Sample Space Definition

A sample space is a set of all possible outcomes of a probability experiment.

Different events for the experiment can be defined, whose outcomes are picked from the sample space. Thus any event defined for the random experiment is a sub set of sample space.

Examples of Sample Space

1. For the most basic experiment of tossing a coin, the sample space consists of only two possible events, turning a head or turning a tail.
S = {H, T}

2. For the experiment of picking a random number from 0 to 9, the sample space consists of 10 possible outcomes as follows:
S = {0,1,2,3,4,5,6,7,8,9}

3. If the question consists of finding the day your friend is available to go for an outing in the coming week, then the sample space would consist of seven outcomes, the days of the week.
S = {Mon, Tue, Wed, Thu, Fri, Sat, Sun}

4. The samples space for a three children family, showing their sexes in the order they were born will contain 8 possible outcomes as shown. The letter B corresponds to a boy, while the letter G indicates a girl.
For the above problem the sample space could also be formed using the number of boys in the family of three children. Here the possible outcomes are 0,1,2 and 3 as the family can have no boys, one boy ,two boys or all three boys.
S = {0,1,2,3}.  But the first method of representing the sample space is better as the outcomes or simple events are equally likely to take place. This is to say that the probability of occurrence is same for all the outcomes in the sample space.

5.  A pack of playing cards consists of 52 cards. For the experiment of picking one card from the pack, the sample space consists of 52 outcomes as shown in the picture below.

Sample Space Example

6.  When two dies are rolled, the expected outcome is an ordered pair. The sample space for the experiment consists of 36 ordered pairs as shown below.
S = { (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
         (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
         (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
         (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
         (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
         (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) }

Sample Space Tree Diagrams

Tree diagrams are used to find and list the sample spaces. The sample space for the gender of the three children in the family is shown below. The branching lines lead to possible successive outcomes at the birth of each child. The final order of outcomes are written after ending the branching line at the birth level of the third child.
Tree Diagrams

Sample Space in Probability

The theoretical probability of an event E can be defined, assuming that all the outcomes in the sample space are equally likely. Equally likely events are events that have the same probability.  For example in the example of picking a card from a pack, all the 52 outcomes in the sample space are equally likely to happen.  This means the probability for each of these simple events = $\frac{1}{52}$
Formula for theoretical probability
Probability of any event E, P(E) = $\frac{Number\ of\ outcomes\ E}{Total\ number\ of\ outcomes\ in\ S}$ = $\frac{n(E)}{n(S)}$ where S is the sample space.
The denominator in the classical probability formula is the number of outcomes in the sample space.
The sample space S can be viewed as an event of occurrence of any one of the all possible outcomes. Hence
P(S) =$\frac{n(E)}{n(S)}$=1. Thus the sample space represents the definite event of an experiment.  If a dice is thrown, we expect any one of the numbers 1,2,3,4,5 or 6 to turn on.  This is definite to happen.  Hence the probability of getting any one of the numbers from the sample space {1,2,3,4,5,6} is 1.

Sample Space Examples

It is not easy to list all the outcomes in the sample space and the event defined in many experiments. But the formula is nevertheless helpful. The following example shows the use of permutation formula to find the number of outcomes in the sample space and to find the probability of the defined event.