2. Using a Sample Space

Overview

A sample space helps you visualize all the possible outcomes of a process. Learn how to create a sample space and use it to calculate probabilities in this lesson.

To explore more Kubicle data literacy subjects, please refer to our full library.

Summary

  1. Lesson Goal (00:14)

    The goal of this lesson is to use a sample space to calculate probabilities.

  2. Viewing the Sample Space (01:08)

    A sample space is the set of all possible outcomes of a process. For example, when we roll two dice, the sample space is the set of all possible combinations of values on the two dice.

    By representing a sample space visually, we can make it easier to identify which possible outcomes correspond to events of interest to us. In our case, we have a 6x6 matrix of outcomes representing the possible values when we roll two dice. We can identify the total number of possible outcomes by calculating the number of items in the matrix.

  3. Calculating Probabilities (02:29)

    To calculate the probability of an event, we divide the number of outcomes where that event occurs by the total number of outcomes. A sample space can help us visualize the outcomes where the event of interest occurs.

    Calculating the probability of different events allows us to compare how likely these events are. Generally if we’re calculating probabilities for different events, it’s useful to express the probabilities as a decimal, as it’s easier to compare decimal values than fractional values.

Transcript

In the previous lesson, we introduced some of the fundamental concepts associated with probability.

In this lesson, we'll learn how to use a sample space to calculate probabilities.

Let's consider a game where we roll two dice and we calculate the sum of the results from both dice.

We'll define three events of interest.

Event A occurs when the total sum of the two dice is greater than or equal to nine.

Event B occurs when the number on the first die is one.

Finally, event C occurs when the number on each die is less than or equal to three.

We want to find the probability of each of these events.

To do this, we need to identify the total number of possible outcomes for the game and the number of outcomes where each event occurs.

One way of doing this is to use a sample space.

The sample space for this game is the set of all possible outcomes of the game.

By representing the sample space visually, we can easily see which outcomes correspond to each of our events.

Let's look at the sample space for this game.

We represent this as a matrix.

The row headings represent the value of the first die.

These possible values range from one to six.

The column headings represent the value of the second die.

They also range from one to six. The values in the body of the matrix represent the sum of the values for the two dice.

For example, if the first die has a value of four and the second die has a value of five, then the total sum is nine. This matrix shows us all the possible outcomes so we can easily identify the number of possible outcomes.

As there are six rows and six columns, this ts six times six, which is 36.

This means there are 36 possible outcomes when we roll two dice.

Let's now calculate the probability for each event.

Event A occurs when the total sum of the two dice is greater than or equal to nine.

We can find the number of outcomes where this occurs by highlighting every outcome with a value of nine or greater.

We can count 10 possible outcomes here.

As a result, the probability of event A is 10/36, which is equal to 0.2778.

This means that when we roll two dice, we'll get a total of nine or more about 28% of the time.

Let's now look at event B.

This event occurs whenever the value on the first die is one.

This event can be represented by all the outcomes of the top row of the matrix.

We can see that there are six possible outcomes here.

Therefore, the probability of this event is 6/36 or 1/6, which is 0.1667.

In this case, it's preferable to express the probability as a decimal, as it makes it easier to compare the probabilities of different events.

By comparing these decimals, we can see that the probability of event B is lower than the probability of event A.

Finally, we'll consider event C.

This event occurs when the value on both dice is three or less.

If we highlight every outcome where this occurs, we can see there are nine possible outcomes.

As a result, the probability of event C is 9/36, which is equal to 1/4 or 0.25.

This tells us that when we roll two dice, we roll a three or less on both dice 1/4 of the time.

As we've seen, using a sample space makes it easy to visually identify all the possible outcomes from a process and identify the outcomes that correspond to events of interest.

In the next lesson, we'll learn how to calculate the probabilities associated with multiple events.