# Probability Fraction Calculator

Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD
Last updated: Jun 05, 2023

Following the definition of probability, we can easily calculate probability as a fraction: with our tool, it will be super easy, barely an inconvenience. If you need to calculate the probability as a fraction for multiple events, you are in the right place! Keep reading for a quick explanation of the math behind the calculations, examples, and applications of the fractional representation of probability.

## What is the probability of an event?

The probability of an event is the measure of the frequency with which said event happens out of a total possible amount of outcomes. If you are dealing with coin tosses, for example, you may find out that head is the result in $495$ out of $1000$ tosses.

There are many ways to express probability, but in general, they all stem from its representation as the ratio between the occurrences of a given outcome and the total number of events happening:

$n_{\mathrm{outcome}}:n_{\mathrm{total}}$

We are used to seeing this ratio expressed as a decimal number, the result of the division of the two members, or as a percentage (the same result, multiplied by $100$. There's, however, an additional way to express probability, and it may come in handy in specific situations: in the next section, we will learn how to calculate probability in fraction form.

## Probability as a fraction

To express probability as a fraction, simply write the number of events that resulted in the desired outcome as the numerator of the fraction and the total number of realizations as the denominator.

You can easily calculate the fraction form of probability with the following formula:

$P(\mathrm{A}) = \frac{n_{\mathrm{A}}}{n_{\mathrm{total}}}$

Where:

• $P(\mathrm{A})$ — The probability of the outcome $\mathrm{A}$;
• $n_{\mathrm{A}}$ — The number of times the event had outcome $\mathrm{A}$; and
• $n_{\mathrm{total}}$ — The total number of events from which we consider the selected outcome.

To calculate the probability as a fraction, follow these steps:

1. Find the number of outcomes and the total number of repetitions.
2. Write the number of outcomes as the numerator and the total number of repetitions as the denominator.
3. Calculate the greatest common divisor of these two numbers. Visit Omni's GCF calculator if you need a refresh on the topic!
4. If the GCF is larger than $1$, divide both the numerator and the denominator of the probability fraction by its value: you will obtain the reduced form of the probability.

## How do I calculate probability as a fraction: the case of multiple events.

In case of multiple outcomes of an event (think of a die and its six faces), we can still calculate the probability as a fraction; however, we need to introduce some small modifications to the process!

Say that you are dealing with an event with possible outcomes $\mathrm{A}$, $\mathrm{B}$, and $\mathrm{C}$. Each of the outcomes happened with the following results:

$\begin{split} \mathrm{A}& \rightarrow n_{\mathrm{A}}\\ \mathrm{B}& \rightarrow n_{\mathrm{B}}\\ \mathrm{C}& \rightarrow n_{\mathrm{C}}\\ \end{split}$

If we sum the occurrences, we find the total number of "realizations":

$n_{\mathrm{total}} = n_{\mathrm{A}}+ n_{\mathrm{B}}+ n_{\mathrm{C}}$

Then, knowing that probability can be a fraction, for the first outcome we write the following expressions:

$P(\mathrm{A}) = \frac{n_{\mathrm{A}}}{n_{\mathrm{total}}}$

For the second and third outcomes, we have, respectively:

$P(\mathrm{B}) = \frac{n_{\mathrm{B}}}{n_{\mathrm{total}}}$

And:

$P(\mathrm{C}) = \frac{n_{\mathrm{C}}}{n_{\mathrm{total}}}$

As a fundamental rule, if the considered outcomes span all the realizations of the event, the following rule holds:

$P(\mathrm{A})+P(\mathrm{B})+P(\mathrm{C})=1$

Calculating the probability of multiple events as a fraction is closely related to another way to represent the same quantities! Think about it: if your outcomes span all the possible events, then their fraction will sum to unity. If we compare unity to a full angle, we can represent the partition of the outcomes as sectors in a pie chart. To understand this comparison even better, visit our three related tools:

With a graphical representation, the analogy will be clear!

## FAQ

### Can probability be a fraction?

Yes: since we define probability as the ratio between the number of events that resulted in a given outcome and the total number of events, we can write these two numbers as the numerator and denominator of a fraction. The fractional representation of probability gives us a quick indication of the magnitude of the probability since its value easily compares to the unit fraction 1.

### How do I find probability in fractions?

To calculate probability in fraction form, follow these easy steps:

1. Find the number of events that resulted in the desired outcome. We call this number nA.
2. Find the total number of realizations, nTOT.
3. Define the fractional form of the probability of the event A as:
P(A) = nA/nTOT
4. Calculate the greater common factor of nA and nTOT. If it's different from 1, divide both numbers by the factor, and find the reduced form of the fraction.

### What is the fraction form of the probability of the results of a coin toss?

1/2 for heads and 1/2 for tails. What do these numbers mean? Take the first fraction:

• The 2 at the denominator is the total number of tosses;
• The 1 at the numerator is the number of tosses resulting in heads.

However, getting two tails or two heads is not so unlikely. Try tossing 1000 times. Heads may be the result of, say, 504 tosses. The fraction 504/1000 is similar in value to 1/2. Repeating the event gives a more accurate fractional representation of the probability.

### How do I calculate the fractional form of the probability of multiple events?

To calculate the fractional form of the probability of multiple events:

1. Define the number of events with defined outcomes: nA, nB, nC, and so on.
2. Calculate the number of total events: nTOT = nA + nB + nC + ...
3. Write the desired number of fractions in the form nA/nTOT, nB/nTOT, nC/nTOT, etc.
4. Simplify the fractions, if possible, using the greater common factor between the respective numerator and the denominator.
Davide Borchia
Enter values in each group
Outcome A
Outcome B
Outcome C
Outcome D
Outcome E
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