In this explainer, we will learn how to interpret a data set by finding and evaluating theoretical probabilities.

Recall that the probability of an event is the likelihood of it happening. The higher the probability of an event, the more likely it is to happen. All probabilities have values in the range from 0 to 1. So, for example, an event with a probability of 0 can never occur (and is therefore impossible), while an event with a probability of 0.9 is very likely to occur. We can express probabilities as fractions or decimals, though sometimes a question will ask for its answer to be given in a specific form.

When working with probability, we often meet activities such as tossing a coin or rolling a die. For instance, we might be asked to find the probability of getting โtailsโ when tossing an unbiased coin or the probability of throwing a 3 when rolling a fair die. Since these are practical activities and are referred to as โexperiments,โ we might think we have to carry them out in real life in order to calculate any probabilities. However, this is not necessarily the case.

Theoretical probability describes the behavior we expect to happen in theory. (This is different from experimental probability, which is obtained through carrying out the same experiment multiple times and analyzing the resulting data. The more times the experiment is performed, the closer the experimental probability approaches the theoretical probability.) To calculate the theoretical probability of an event, we need to be given a precise description of the system (experiment) that gives rise to it. We then use logic to analyze the system and work out how it should behave in theory.

First, we recall some basic terminology relating to experiments.

### Definition: Outcomes, Sample Spaces, and Events

The possible results of an experiment are called outcomes (or occurrences).

The sample space is the set of all possible outcomes.

An event is a set of outcomes to which a probability is assigned. It is a subset of the sample space , written . We use the term โfavorable outcomesโ to refer to the outcomes we want to test for in an experiment.

For a given experiment, once we have identified the sample space and the event we want to test for, we can calculate the theoretical probability of that event using the following formula.

### Formula: Theoretical Probability

More formally, let be the sample space, be the event, be the number of elements in , and be the number of elements in . Then, the probability of the event, , is given by the formula

The formula is stated in two ways: a worded version and a symbolic version. Often, when faced with a specific question, we may find one version more natural to use than the other, but, of course, both will give the same result.

Note that this formula only works for fair experiments; this means that there must be no bias involved so that all of the individual outcomes are equally likely to occur. This lack of bias is highlighted in the wording of questions by the phrases describing items featured in an experiment (e.g., โfair dieโ or โunbiased coinโ) and the manner in which these items are selected (e.g., โpicked at randomโ or โrandomly chosenโ).

To summarize, if given sufficient information about the properties of a fair experiment, we can always calculate the theoretical probability of an event by applying the following procedure.

### How To: Calculating the Theoretical Probability

- Identify all the possible outcomes of the experiment and count how many there are. This number is .
- Identify the outcomes we want to measure (i.e., the favorable outcomes) and count how many there are. This number is .
- Divide by to get the theoretical probability.

We are now ready to answer some questions on this topic.

### Example 1: Determining the Theoretical Probability of an Event Given as a Word Problem

A box contains 5 red marbles, 8 green marbles, and 4 yellow marbles. If a marble is picked from the box at random, what is the probability that the marble is red?

### Answer

Recall that we must identify all the possible outcomes (the sample space) and the ones we want to measure (the favorable outcomes). Once we know the number of elements in each of these sets, we can apply the theoretical probability formula to find the probability of the event.

First, consider all the possible outcomes. The experiment involves a box of marbles from which one is picked at random. The possible outcomes are the individual marbles, and so the sample space is the set of all the marbles in the box. Therefore, the number of elements in the sample space is the total number of marbles. We are told that 5 marbles are red, 8 are green, and 4 are yellow, giving a total of

Next, from the question, the event we want to measure is picking a red marble. Therefore, the favorable outcomes are the red marbles in the box, and we know that there are 5 of these.

Finally, we can apply the formula as follows. which is a fraction in its simplest form. Note that it is preferable to give this answer as an exact fraction rather than converting it to the nonterminating decimal and rounding.

We conclude that if a marble is picked from the box at random, the probability it is red is .

In the above example, the event we wanted to measure involved a single type of outcome, namely, picking a marble of one color from the box. Sometimes, however, the event may involve more than one type of outcome.

### Example 2: Determining the Theoretical Probability of an Event in a Dice Experiment

What is the probability of rolling a number greater than or equal to 4 on a fair die

### Answer

Recall that we must identify all the possible outcomes (the sample space ) and the ones we want to measure (the favorable outcomes ). Once we know the respective numbers of elements in these sets, and , we can apply the theoretical probability formula to find .

When rolling a fair die, the possible outcomes are the scores on each of the six faces, so . The number of elements in this set is

To roll a number greater than or equal to 4, we must roll a 4, 5, or 6. Therefore, , and the number of elements in this set is

Then, we can apply the formula, so

Note that we can convert this exact fraction answer to the exact decimal if needed.

Thus, the probability of rolling a number greater than or equal to 4 is 0.5.

Examining the right-hand side of the formula , we can see that another way to think of the theoretical probability of an event is as the proportion of the total number of outcomes that are favorable. Alternatively, we could express this as the fraction of the sample space taken up by the outcomes that make up the event. This means that sometimes we can still calculate the theoretical probability of an event without knowing the size of the sample space.

### Example 3: Determining the Theoretical Probability of an Event Involving Colored Balls

A bag contains an unknown number of balls. Given that one-sixth of the balls are white, one-fifth of them are green, and the rest are blue, what is the probability that a ball drawn at random from the bag is blue?

### Answer

In this question, we are given the fractions corresponding to the probabilities that a ball drawn at random from the bag is either white or green (i.e., two of the three possible sets of outcomes of the total sample space). We need to find the probability of the one remaining set of outcomes: that a ball drawn at random from the bag is blue.

The sample space is all of the balls in the bag, the total number of which is unknown.

When drawing a ball at random, there are three separate possible events as follows.

- Event : the ball is white. We write for the probability of this event.
- Event : the ball is green. We write for the probability of this event.
- Event : the ball is blue. We write for the probability of this event.

From the wording of the question, we can read off these values:

We need to work out , which is the missing fraction of the sample space. Recall that the sum of the probabilities of all the outcomes in a sample space has to equal 1. Therefore, we have

To make the subject of this formula, subtracting and from both sides gives

Now, we substitute the values of and to get

We can simplify the right-hand side by writing all three terms as fractions with a common denominator of 30 (the lowest common multiple of 6 and 5). Thus, we have which is a fraction in its simplest form.

Therefore, the probability that a ball drawn at random from the bag is blue is .

In some questions, rather than calculating probabilities of events, we are given the probability and have to work backward using the formula to find the size of the set of outcomes corresponding to a particular event.

### Example 4: Determining the Number of Marbles in a Bag Using Theoretical Probability

A bag contains 30 colored marbles. The probability of choosing a white marble at random is . How many white marbles are in the bag?

### Answer

Here, we are given the sample space and its size . We also know the event we are interested in, , and its theoretical probability, . Our strategy will be to substitute these values into the formula for theoretical probability and work backward to find .

The sample space is the set of all the marbles in the bag. The number of elements in this set is

The required event is choosing a white marble at random from the bag. From the question, we know the theoretical probability of this event, so

Since we need to find the number of white marbles , we rearrange the formula to make this the subject. Starting with we multiply both sides by to give

Now, we can substitute the values of and from above, so

Therefore, we have shown that there are 12 white marbles in the bag.

Some theoretical probability problems contain extra details that are not needed for the solution. It is always important to read each question carefully to extract the relevant information.

### Example 5: Determining the Theoretical Probability of an Event Involving Passing or Failing a Test

In a class of 50 students, 33 passed the mathematics test and 31 passed the language test. What is the probability that a randomly selected student failed the language test?

### Answer

Recall that we must identify the sample space and the event we want to calculate the probability of, . Once we know the respective numbers of elements in these sets, and , we can apply the theoretical probability formula to find .

The sample space is the entire class of students; the number is

The required event is failing the language test. This means we must focus on the language test, so the information given in the question about the mathematics test is not relevant to solving this problem. We are not told how many students failed the language test, but we do know how many passed, which is 31. Hence, we can work out by subtracting the number of students who passed the language test from the total number in the class. Thus, we have

Finally, we apply the formula to get which is a fraction in its simplest form. Note that we can convert this exact fraction to the exact decimal .

Therefore, the probability that a randomly selected student failed the language test is 0.38.

Let us finish by recapping some key concepts from this explainer.

### Key Points

- When given a description of an experiment, we need to identify its sample space (the set of all possible outcomes) and the event we want to measure (the set of favorable outcomes). Then, the theoretical probability of the event is given by the following formula: More formally, let be the sample space, be the event, be the number of elements in , and be the number of elements in . Then, the probability of the event, , is given by the formula
- We can work backward from a given probability using the formula to find the size of the set of outcomes corresponding to a particular event.
- Since the theoretical probability of an event is also the proportion of the total number of outcomes that are favorable, sometimes we can calculate a missing theoretical probability even if we do not know the size of the sample space.