# Probability Fraction Calculator

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:

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:

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:

- Find the number of outcomes and the total number of repetitions.
- Write the number of outcomes as the numerator and the total number of repetitions as the denominator.
**Calculate the greatest common divisor**of these two numbers. Visit Omni's GCF calculator if you need a refresh on the topic!- 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:

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

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

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

And:

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

## Other related tools:

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:

- Find the number of events that resulted in the desired outcome. We call this number
`nA`

. - Find the total number of realizations,
`nTOT`

. - Define the fractional form of the probability of the event
`A`

as:

`P(A) = nA/nTOT`

- 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:

- Define the number of events with defined outcomes:
`nA`

,`nB`

,`nC`

, and so on. - Calculate the number of total events:
`nTOT = nA + nB + nC + ...`

- Write the desired number of fractions in the form
`nA/nTOT`

,`nB/nTOT`

,`nC/nTOT`

, etc. - Simplify the fractions, if possible, using the greater common factor between the respective numerator and the denominator.