Probability Using Sample Spaces


What is a Sample Space of an Experiment?

A sample space is simply all the possible outcomes of an experiment.

For example, if you flip a coin once the sample space is {heads, tails}.

If you flip a coin twice in a row, the sample space is {(heads, heads), (heads, tails), (tails, heads), (tails, tails)}.

When referring to a sample space, we write the list of possible outcomes in curly brackets {}.

Let’s walk through some examples of finding probability using sample spaces.

Example 1: You flip two coins at once. What is the probability that both coins land on tails?

Recall that we find the probability of some event “A” happening by using the following formula:

The sample space describes the total number of outcomes. In this case, the possible outcomes when we flip two coins is:

{(heads, heads), (heads, tails), (tails, heads), (tails, tails)}

So there are four possible outcomes. And both coins land on tails in only one of those outcomes:

{(heads, heads), (heads, tails), (tails, heads), (tails, tails)}

Thus, the probability that both coins land on tails is 25%:


Example 2: You flip two coins at once. What is the probability that at least one coin lands on heads?

First, find the sample space:

{(heads, heads), (heads, tails), (tails, heads), (tails, tails)}

Next, find all of the outcomes that include at least one heads. It turns out there are three:

{(heads, heads), (heads, tails), (tails, heads), (tails, tails)}

Thus, the probability that at least one coin lands on heads is 75%:


Example 3: You flip three coins at once. What is the probability that none of them land on heads?

First, find the sample space: (“H” means “heads”, “T” means “tails)

{(HHH), (HHT), (HTH), (HTT), (THH), (THT),  (TTH), (TTT)}

In this case, there are eight possible outcomes when we flip three coins.

Next, find all of the outcomes that include no heads. It turns out there is only one (the outcome in which all three coins land on tails):

{(HHH), (HHT), (HTH), (HTT), (THH), (THT),  (TTH), (TTT)}

Thus, the probability that none of the coins land on heads is 12.5%:

Example 4: You roll a die twice. What is the probability that it lands on the same number on both rolls?

First, find the sample space. In this case, it’s helpful to use a grid to represent all the possible outcomes:

Next, find every possible outcome where both dice land on the same number. It turns out that there are six outcomes where this could happen:

Thus, the probability that the die lands on the same number on  both rolls is 1/6:

Example 4: You roll a die and flip a coin at the same time. What is the probability that the coin lands on heads and the die lands on an even number?

First, find the sample space. Again it’s helpful to use a grid to represent all the possible outcomes:


Next, find every possible outcome where the coin lands on heads and the die lands on an even number. There are three outcomes where this could happen:


Thus, the probability that the coin lands on heads and the die lands on an even number is 25%:

Leave a Reply

Your email address will not be published. Required fields are marked *