Probability is useful in everyday life, from guessing the number of jellybeans in a jar to anticipating the weather. That’s why I’ve decided to teach you everything you need to know about the **theory of probability**, from the basic concepts to more complex examples. By the end of this tutorial, you will know how to calculate probability like a pro!

Contents

- What is Probability?
- History of Probability
- Probability Formula
- How to Calculate the Probability of a Single Event
- Now let us try it for all three examples!
- How To Calculate Probability – Three Types of Probability
- Vocabulary
- Probability Distributions
- The Future of Probability
- How To Calculate Probability – Final Thoughts

## What is Probability?

Probability (or P) is the **likelihood of an event occurring**. It is a measure of how often we expect something to happen. It can be expressed as a number between 0 and 1, where 0 means an event will never happen, and 1 means an event will definitely happen. For example, the P value of flipping a coin and it lands on heads is 0.5 or 50%.

Read more: What Does a Meteorologist Do?

## History of Probability

The **concept of likelihood** has been around for centuries. It was first studied by mathematicians in the 16th century as a way to **analyze games of chance**. Probability theory was formalized in the 19th century by French mathematician Pierre-Simon Laplace. Since then, it has been used in a wide variety of applications, from **predicting the stock market** to understanding DNA.

## Probability Formula

The probability of an event happening is calculated by:

P(event) = Number of ways the event can happen / Total number of possible outcomes

For example, let us say we have a jar of 20 marbles: 10 red, five blue, and five green. If we pick a marble out of the jar at random, what is the likelihood of it being red?

There are ten ways the event (picking a red marble) can happen. There are 20 total possible outcomes (picking any marble). So, the probability of picking a red marble is 10/20 or 0.5.

## How to Calculate the Probability of a Single Event

Now that we know the formula for probability, let us look at some examples.

### Example 1

What is the likelihood of flipping a coin and it landing on tails?

There are two possible results when flipping a coin: heads or tails. So, the probability of flipping a coin and it lands on tails is 1/2 or 0.5.

### Example 2

What is the probability of rolling a die and it landing on a 4?

There are six possible outcomes when rolling a die: 1, 2, 3, 4, 5, or 6. So the probability of rolling a die and it lands on a 4 is 1/6 or 0.167.

### Example 3

Let us make it a bit more difficult. What is the likelihood of drawing an Ace from a deck of cards?

Well, there are 52 cards in a deck: 4 Aces, 4 Kings, 4 Queens, and so on. There are 13 cards of each type (suit). So, the probability of drawing an Ace is 4/52 or 0.077.

### How to Calculate the Probability with Multiple Random Events

Sometimes we want to know the chance of multiple events happening at the same time. For example, what is the chance of flipping a coin and it landing on tails AND rolling a die and it landing on a 4? To calculate this, we use the multiplication rule.

The multiplication rule says that the P value of two events happening at the same time is:

### P(A and B) = P(A) x P(B)

where A and B are two events.

There are two potential endings when flipping a coin: heads or tails. There are six possible outcomes when rolling a die: 1, 2, 3, 4, 5, or 6. So the likelihood of flipping a coin and it lands on tails AND rolling a die and it lands on a 4 is (1/2) x (1/6) or 0.083.

## Now let us try it for all three examples!

What is the probability of flipping a coin and it landing on tails AND rolling a die and it landing on a 4 AND drawing an Ace from a deck of cards?

P (A, B, and C) = P(A) x P(B) x P(C)

= (1/2) x (1/6) x (4/52)

= 0.0064

### Odds vs. Probability

You may have heard the terms “odds” and “probability” used interchangeably, but they are actually quite different. Odds are a ratio of favorable outcomes to unfavorable outcomes. Probability is a measure of how often we expect something to happen.

### To calculate odds, we use the following formula:

Odds = Number of favorable outcomes / Number of unfavorable outcomes

For example, let us say we want to know the odds of flipping a coin and it lands on tails. There are two possible outcomes when flipping a coin: heads or tails. So, the odds of flipping a coin and it lands on tails is 1/1 or 1 to 1.

If we make this a bit more complicated, we can use our Ace example from earlier. What are the odds of drawing an Ace from a deck of cards?

There are four Aces and 52 total cards in a deck. So, the odds of drawing an Ace from a deck of cards is 4/48 or 4 to 48. You would then round this down to 1 to 12 as you want your odds to be in its lowest possible form.

## How To Calculate Probability – Three Types of Probability

Okay, so **basic probability** has been covered, but from here, things get a little bit more complicated. There are three types of it: Classical, Relative Frequency, and Subjective Probability.

### Classical Probability

This is what we have been talking about so far. It is based on the idea of equally likely outcomes. This means that each outcome has the same chance of happening.

### Relative Frequency Probability

This is based on the idea of actual outcomes. For example, if we flip a coin 100 times and it lands on tails 50 times, then the relative frequency probability of flipping a coin, and it landing on tails would be 50%.

### Subjective Probability

This is based on an individual’s personal judgment. For example, if I think that there is a 90% chance of it raining tomorrow, then the subjective probability of it raining tomorrow is 90%.

Classical and relative frequency probabilities can be calculated using mathematics. Subjective probability cannot be calculated as it is based on someone’s personal opinion.

## Vocabulary

Let’s now review some important vocabulary…

Random Event – An event that cannot be predicted with certainty. For example, flipping a coin.

Outcome – The result of a random event. For example, the outcome of flipping a coin is either heads or tails.

Sample Space – The set of all feasible results for a random event. For example, the sample space for flipping a coin is {heads, tails}.

Event – A subset of the sample space. For example, the event “flipping a coin and it lands on tails” is {tails}.

Probability – A measure of how often we expect something to happen. Can be either classical probability, relative frequency probability, or subjective probability.

## Probability Distributions

There is one more topic that I need to cover before we can start calculating probabilities, and that’s probability distributions. This is a mathematical function that describes the likelihood of a random event occurring.

There are two types of probability distributions: Discrete and Continuous.

### Discrete Probability Distribution

This is used when the outcomes of a random event can be listed out. For example, the likelihood of flipping a coin and it lands on either heads or tails can be described using a discrete probability distribution.

### Continuous Probability Distribution

This is used when the outcomes of a random event cannot be listed out. For example, the chance of a person being between the heights of 5’0″ and 6’0″ can be described using a continuous probability distribution.

## The Future of Probability

Now that you know how to calculate it, you are well on your way to understanding statistics. This theory is a **critical part of statistics**, and it can be used to help **make predictions about the future**. With a little practice, you will be able to use it to your advantage in many different situations.

It also forms the basis of the following complex statistical tests:

- T-Tests
- Z-Tests
- F-Tests
- Chi-square tests
- Occupancy Modelling
- Monte Carlo Simulation
- …and many more!

Out with data science, it can be part of many jobs; for example:

- Calculating weather predictions as a meteorologist
- Figuring out if a medicine works as a doctor
- Predicting demand as a business analyst

There are many other examples, but these should give you an idea of how important it is in the real world.

### Want to learn more about the fascinating subject of Probability?

Then take a look at Probability: For the Enthusiastic Beginner, PROBABILITY: Calculate Your Chance: Substantiate Your Decision, or the excellent Probability For Dummies and Probability Demystified, all available online in 2023.

And for more focused uses of probability, you’ll love LIFE HACKS AND THE NUMBERS: Calculate your chances to success, Introduction to Probability with Texas Hold ’em Examples, High Probability Trading: Take the Steps to Become a Successful Trader, Advanced Techniques in Day Trading: A Practical Guide to High Probability Strategies and Methods and How Professional Traders in an Investment Bank calculate Probability in the stock market.

## How To Calculate Probability – Final Thoughts

If you made it this far, then congratulations, you now know the **best way to calculate probability**!

This theory is a vital component of statistics, and it is a great tool to have in your data science arsenal. With a little practice, you will be able to use likelihood to your advantage in many different situations, so get ready to place your bets!

Happy calculations!