# Lesson Explainer: Introduction to Probability Mathematics

In this explainer, we will learn how to use probability terms and axioms to understand the concept of probability.

We know when we speak of probability that we are referring to the chance or likelihood of an event occurring.

Theoretical probability is what we expect to happen based on mathematical reasoning. For example, if we toss a fair coin, we know that

1. there are two possible outcomes: the coin will land with either heads or tails showing upward;
2. theoretically, each of these has a 50% (or a 1-in-2) chance of occurring.

Let us look at another example.

### Example 1: Theoretical Probability with a Fair Die

What is the probability of rolling an even number on a fair die?

To find the probability of rolling an even number on a fair die, we note that there are 6 faces on a die:

and that 3 of those 6 faces (2, 4, and 6, highlighted in red) show an even number of dots:

The probability of rolling an even number is, therefore, . As a decimal, dividing 1 by 2, this is 0.5. And as a percentage, there is a 50% chance of rolling an even number ().

We have used a number of the concepts of probability in this example; so before continuing with some more examples, let us recall some of these ideas and what we mean by theoretical probability.

### Some Probability Concepts and Theoretical Probability: Main Points

In probability theory, we have the following.

• An experiment is a specific procedure which can be repeated as many times as we like and which has the same set of randomly occurring possible results each time.
• An outcome is a specific result of an experiment.
• The sample space, , of an experiment is the set of all possible outcomes. The sample space is an exhaustive set; hence, the sum of the probabilities for all possible outcomes in the set is equal to 1.
• An event is a subset of the sample space, containing one or more possible outcomes of the experiment.
• The theoretical probability of an event is the number of ways the event can occur (favorable outcomes) divided by the total number of possible outcomes:
• All probabilities must be between 0 and 1 (inclusive); that is,. An event with probability zero cannot happen (i.e., it is an impossible event), whereas an event with probability 1 is certain to occur. As the probability of an event gets closer to 1, it is more likely to occur; and as the probability gets closer to zero, the event is less likely to occur. It is useful to illustrate this on a probability scale.

Note also that probabilities can be written as either decimals, fractions (between 0 and 1), or percentages (0% to 100%).

Applying these ideas to the probability of throwing an even number on a fair die, the experiment is “throwing a fair die,” and there are six possible outcomes: 1, 2, 3, 4, 5, and 6. Hence, the sample space . Each of these outcomes has a 1-in-6 chance of occurring (i.e., a probability of ). If we let “throwing an even number” be event , then has three favorable outcomes: . The probability of throwing an even number is, therefore, .

The next three examples test our understanding of probabilities.

### Example 2: Possible Probabilities

Which of the following choices may represent the probability of an event occurring?

1. 197%
2. 1.3
3. 14%

Let us consider each option in turn to determine which one may represent the probability of an event occurring. Option (A) is 197%. In percentage terms, we know from the rules of probability that all probabilities must be between 0% and 100% (inclusive). 197% is not between 0% and 100%; hence, 197% cannot represent the probability of an event occurring.

Option (B) is 1.3. In terms of decimals, we know from the rules of probability that, for any event , . Since 1.3 is greater than 1, 1.3 cannot represent the probability of an event occurring.

Option (C) is . Again, in terms of decimals, since is a negative number, it does not lie between 0 and 1. Hence, cannot represent the probability of an event occurring.

Our final option is option (D), which is 14%. Since 14% does lie between 0% and 100%, 14% could represent the probability of an event occurring.

Hence, option (D) (14%) is the only option that could represent the probability of an event occurring.

### Example 3: Possible Probabilities

What is the probability of a family visiting the beach in a given year?

1. 0
2. 1
3. Between 0 and 1

If an event has probability equal to 0, then it is an impossible event and cannot occur. We do not have enough information to state for certain that a family will definitely not visit the beach in a given year, so we cannot choose option (A) (i.e., 0) as our answer.

If an event has probability equal to 1, it is a certain event; that is, it will definitely occur. Since we do not have enough information to state for certain that the family will definitely visit the beach in a given year, we cannot choose option B (i.e., 1) as our answer.

We do, however, know that all probabilities must lie between 0 and 1 (inclusive, i.e., in the set containing 0, 1, and everything between them). So the probability that a family visits the beach in a given year must be somewhere between 0 and 1 (inclusive). Hence, we must choose option (C) as our answer.

### Example 4: Possible Probabilities

What is the probability of rolling a number divisible by 7 on a regular die?

There are 6 faces on a fair die, numbered 1 to 6.

The probability of rolling a number divisible by 7 on a regular die is the number of face values of a fair die divisible by 7 divided by the total number of face values. That is,

Since all the face values on a regular die are less than 7, they cannot be divisible by 7. So the number of face values divisible by 7 is equal to 0. Hence,

The probability of rolling a number divisible by 7 on a regular die is 0.

In our next two examples, we will work out theoretical probabilities involving the sections of a fair spinner.

### Example 5: Theoretical Probabilities for a Fair Spinner

What is the probability that the pointer lands on a number greater than 11 when the given fair spinner is spun?

To find the probability that the pointer lands on a number greater than 11, we need to know (1) how many of the numbers on the spinner are greater than 11 and (2) how many numbers there are on the spinner, that is, how many sections the spinner has in total.

We can see that the numbers in all of the sections in the bottom half of the spinner are greater than 11:

and that there are 4 of these, numbered 12, 13, 14, and 15. We can also see that there are 8 sections in total (2 blue, 2 red, 2 yellow, and 2 green sections). Hence, the probability that the pointer lands on a number greater than 11 is

Hence, the probability that the spinner lands on a number greater than 11 is , which is 0.5 as a decimal. Or, converting to a percentage (), we can say that there is a 50% chance the spinner lands on a number greater than 11.

### Example 6: Theoretical Probabilities for a Fair Spinner

What is the probability of the pointer landing on a prime number when the given spinner is spun?

To find the probability of the pointer landing on a prime number when the given spinner is spun, we need to know (1) how many of the numbers on the spinner are prime numbers and (2) how many numbers (or sections) there are on the spinner in total.

Noting that a prime number is divisible only by 1 and itself, and assuming that 1 is not a prime number, we can see that there are 3 sections on the spinner with prime numbers: those numbered 2, 7, and 11.

Counting the number of sections in total, we have 8 sections (2 blue, 2 red, 2 yellow, and 2 green sections).

Hence,

The probability that the pointer lands on a prime number is therefore 0.375 or, as a percentage, ; that is, there is a 37.5% chance that the pointer lands on a prime number.

Finally, let us remind ourselves of the main points on theoretical probability.

### Key Points

• The theoretical probability of an event is the number of ways the event can occur (favorable outcomes) divided by the total number of possible outcomes:
• All probabilities must be between 0 and 1 (inclusive); that is,. An event with probability zero cannot happen (i.e., it is an impossible event), whereas an event with probability 1 is certain to occur. As the probability of an event gets closer to 1, it is more likely to occur; and as the probability gets closer to zero, the event is less likely to occur. It is useful to illustrate this on a probability scale.

Note also that probabilities can be written as either decimals, fractions (between 0 and 1), or percentages (0% to 100%).