In this explainer, we will learn how to find the probability of a simple event.

First, letβs recall the meaning of probability. Probability is the likelihood or chance of an event occurring. All probabilities lie in the interval , and the probability that an event is:

- certain is 1,
- impossible is 0,
- likely is closer to 1 than 0,
- unlikely is closer to 0 than 1,
- equally likely as unlikely is .

We can represent the events and their probabilities by using a number line, as shown below.

In order to calculate the probability of an event occurring, we need to know the number of favorable outcomes and the number of total outcomes. The probability is then calculated by dividing the number of favorable outcomes by the total number of outcomes. In other words,

This is defined more formally below.

### Definition: The Probability of an Event

If is an event in a sample space , then the probability of event occurring is where represents the probability of event , represents the number of elements in event , and represents the number of elements in sample space , where each element has an equally likely outcome.

From this definition, we can show that the probability of event must lie in the interval .

Since ( is a subset of ), and since and ,

Therefore,

In the first example, we will discuss how to find the probability of an event where there are only two possible outcomes.

### Example 1: Determining the Theoretical Probability of an Event

A class contains 6 boys and 21 girls. What is the probability of selecting a girl if a student is selected at random?

### Answer

To find the probability that a randomly selected student is a girl, we use the following formula to find the probability of an event: where represents the probability of event , represents the number of elements in event , and represents the number of elements in sample space .

In this question, event represents the event that if a student is selected at random, then they are a girl, and sample space represents the boys and girls in the class.

To find the probability of event , we need to find both the number of elements in , and the number of elements in sample space .

We are told that the number of girls in the class is 21, so the number of elements in is .

To find , the number of elements in the sample space, we add the number of girls and the number of boys together, as these are the only given possible outcomes. This gives us students. So, the number of elements in the sample space is .

Using the formula above, we can find . By substituting and , we get

Therefore, the probability that a student selected is a girl is .

In the next example, we will find the probability of an event as we did in the previous example, but this time there are three possible outcomes.

### Example 2: Determining the Theoretical Probability of an Event in a Marbles Problem

A bag contains 7 white marbles, 8 black marbles, and 7 red marbles. If a marble is chosen at random from the bag, what is the probability that it is white?

### Answer

To find the probability that a randomly chosen marble is white, we use the following formula to find the probability of an event: where represents the probability of event , represents the number of elements in event , and represents the number of elements in sample space .

In this question, event represents the event that a white marble is chosen from the bag, and sample space represents all of the white, black, and red marbles in the bag.

To find the probability of event , we need to find both the number of elements in , and the number of elements in sample space .

We know there are 7 marbles in the bag, so the number of elements in is .

To find the number of elements in sample space , we need to add the number of white marbles, the number of black marbles, and the number of red marbles, as these are the only possible outcomes. Summing these together, we get marbles. So, the number of elements in the sample space is .

Using the formula above, we can find . By substituting and , we get

Therefore, the probability that a marble selected from the bag is white is .

So far, we have considered examples where we are given the number of occurrences of an event. Next, we will consider an example where we need to find the number of occurrences of an event given the description of the event.

### Example 3: Finding the Probability of Rolling an Odd Number on a Fair Die

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

### Answer

To determine the probability of rolling an odd number on a fair die, we use the following formula to find the probability of an event: where represents the probability of event , represents the number of elements in event , and represents the number of elements in sample space .

In this question, event represents the event that the die shows an odd number, and sample space represents all the possible numbers on the die.

To find the probability of event , we need to find both the number of elements in , and the number of elements in sample space .

We know that on a fair die there are 6 possible outcomes: 1, 2, 3, 4, 5, and 6. Therefore, the number of elements in sample space is 6. Of these outcomes, 1, 3, and 5 are odd, so the number of elements in event is 3.

Therefore, we can say that and .

Using the formula above, we can find . By substituting and , we get

Therefore, the probability of getting an odd number on a die is .

Next, we will discuss an example where we need to determine the number of favorable outcomes in order to calculate the probability of an event.

### Example 4: Determining the Probability of an Event

What is the probability of selecting at random a prime number from the numbers 8, 9, 20, 19, 3, and 15?

### Answer

To find the probability of selecting a prime number from the numbers 8, 9, 20, 19, 3, and 15, we use the probability formula for an event, as follows: where represents the probability of event , represents the number of elements in event , and represents the number of elements in sample space .

Here, event represents the event that a prime number is chosen, and sample space represents the list of numbers 8, 9, 20, 19, 3, and 15.

In order to calculate the probability, we need to find the number of elements in event , , and the number of elements in sample space , .

As the numbers 8, 9, 20, 19, 3, and 15 represent sample space , the number of elements in is 6.

From the list of numbers 8, 9, 20, 19, 3, and 15, the prime numbers (those with exactly 2 factors) are 19 and 3. Therefore, the number of elements in event is 2.

Therefore, we can say that and .

Using the formula above and substituting and , we get

Therefore, the probability of selecting a prime number from the list of numbers 8, 9, 20, 19, 3, and 15 is .

In the following example, we will discuss how to find the number of elements in the sample given the number of elements in the event and the probability of a different event.

### Example 5: Using Theoretical Probability to Solve a Problem

A bag contains 24 white balls and an unknown number of red balls. The probability of choosing at random a red ball is . How many balls are there in the bag?

### Answer

In this question, since we are working with the probability of selecting a red ball and need to find the total number of balls in the bag, we need to use the following formula for the probability of an event: where represents the probability of event , represents the number of elements in event , and represents the number of elements in sample space .

Here, we are given the probability of selecting a red ball, and the total number of white balls, so it is easier to work with the event in which we have the probability, which is the event that we select a red ball. So letβs define event as the event of selecting a red ball and sample space as all the balls in the bag.

We are told that the bag contains both red balls and white balls. So, the number of elements in sample space is the sum of 24 white balls and an unknown number of red balls. If we let represent the number of red balls, then we can say the number of elements in the sample space is

As we have set representing the number of red balls, the number of elements in event is .

We are told that the probability of choosing a red ball is , so the probability of event is .

Since we have , , and , we can substitute these into the formula above to find and, therefore, , the number of balls in the bag:

Rearranging and solving for , we get

Therefore, the number of red balls in the bag is 7.

We can now find the number of balls in the bag, since the number of white balls in the bag is 24 and the number of red balls is 7, meaning the total number of balls is .

Therefore, the total number of balls in the bag is 31.

### Note

You may have noticed that the denominator of the probability of getting a red ball, , is the same as the total number of balls in the bag. Since the denominator in the formula for finding the probability of an event is the total number of elements in the sample space, then it is likely that the denominator in the probability of getting a red ball is the total number of balls. However, if the fraction has been simplified, then the denominator would be a factor of the total number of balls.

In the last example, we will discuss how to find the total number of members of the sample space, given the probabilities of two events and the number of occurrences of a third event.

### Example 6: Using Simple Probability to Solve a Problem

A bag contains an unknown number of marbles. There are 3 red marbles, some white marbles, and some black marbles. The probability of getting a white marble is , and the probability of getting a black marble is . Calculate the number of marbles in the bag.

### Answer

In this question, since we are working with the probability of getting a white marble and the probability of getting a black marble, we use the formula for finding the probability of an event, as follows: where represents the probability of event , represents the number of elements in event , and represents the number of elements in sample space .

Instead of saying event , letβs say the event that a white ball is selected is denoted by W, the event that a black ball is selected is denoted by B, and the event that a red ball is selected is denoted by R. Sample space here is all the balls in the bag.

We are told in the question that the number of red marbles is 3, so the number of elements in event R is .

As we do not know the number of white marbles nor the number of black marbles, letβs say the number of white marbles is w and the number of black marbles is b. So, and .

Since the only possible outcomes are a red marble, a white marble, or a black marble, the number of elements in the sample space is the number of red marbles plus the number of white marbles plus the number of black marbles.

We know the number of red marbles is 3 and have defined the number of white marbles as w and the number of black marbles as b, so the number of elements in sample space is

We are told that the probability of selecting a white marble is , and since we have defined the number of white marbles as and have found that the number of balls in the bag is , we can substitute these into the formula

This gives us

Rearranging by cross-multiplying by the denominators, we get

Subtracting w on both sides, we get

We will call this equation 1.

Similarly, we are told that the probability of selecting a black marble is , so since we have defined and have found , we can substitute these into the formula

This gives us

Rearranging by cross-multiplying by the denominators, we get

Subtracting b on both sides, we get

We will call this equation 2.

Since we have two equations with two unknowns, we can solve for w and b using simultaneous equations.

We set equation 1 as and equation 2 as .

We can solve these by substituting b in equation 1 for from equation 2. This gives us

So, the number of white marbles is 6.

If we substitute into equation 2, we will get

So, the number of black marbles is 9.

To find the number of marbles in the bag, we can add the number of red marbles, which we know from the question is 3, and the number of white marbles, which we found to be 6, and the number of black marbles, which we found to be 9. This gives us

Therefore, the number of marbles in the bag is 18.

### Note

We could have alternatively solved this by finding the probability of getting a red marble first using the fact that all probabilities add up to 1.

In this explainer, we have learned how to find the probability of an event occurring and work with problems where we needed to find the number of favorable outcomes and the total number of outcomes, as well as solve problems where the probability is given.

### Key Points

- We can describe the probability of an event as
- All probabilities lie in the interval , with a probability closer to 1 being more likely and a probability closer to 0 being less likely.
- Formally, we say the probability of an event is where represents the probability of event , represents the number of elements in event , and represents the number of elements in sample space .