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Introduction to Probability and Statistics

Introduction to Probability and Statistics -magoosh

How likely are you to flip a coin and it lands on its edge? How likely is that you wear a blue shirt today? Best of all, how likely is it that I can taste the difference between coffee poured into cream versus cream poured into coffee? Provided we have the right information, probability and statistics can tell us the those likelihoods.


The likelihood of something happening is the probability of that thing, called and event, happening. Let’s take a look at a couple of typical examples.

When you roll a D20 in the game of Dungeons & Dragons, getting a 20 is a very good thing most of the time. Of course, everyone talks about this like it is super rare when, in reality, it is just as likely to roll a 20 as a 1. Let’s figure out why. I’ll be using set notation, so if you need a review, check this out.

When we are trying to figure out probability (P), you are trying to figure out the chance of an event occurring. The probability of an event occurring is usually written as P(event). In the case of our D&D dice when you roll it you have these outcomes

S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

This means that the likelihood of rolling a number 1 – 20 is 100% while the likelihood of rolling a 21 is 0%. But the events in between are a little different.

Probability is calculated as the total number of desired outcomes ÷ total number of possible outcomes.

Our Example

There are 20 total possible evens that can occur on a single roll of a fair D20. At this time, we are only interested in one of them, 20. Since it is one of 20 possible outcomes, P(20) = 1/20 = 0.05. It has the same likelihood of a 1 being rolled, P(1) = 1/20 = 0.05.

Now, rolling a number less than twenty is different. This would be

P(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19) = 19/20 = 0.95, which is much more likely.

So, rolling a D20 and getting a 20 is just as likely as rolling a 1. However, the reason 20s are so rare is that you are much more likely to roll a number less than 20. On a side note, check this out for more info on this type of event.


Now that we have a general idea of probability, let’s discuss probability in practice: statistics. Statistics is the application of the laws of probability to actual data. If we take the D20 example, this would be when you roll the dice 20 times and collect some data.

When we apply probability to real data, we are trying to determine if the outcome is significantly different from a model that we are generating. For example, the P(20) = 0.05, so let’s explore that.

When you collect data, there are several ways to describe the data that you take. The most common are mean, median, and mode. In the case of statistics, we want to see if our actual data conforms to the model. There are two ways to do this: classical inference and Bayesian inference.

Classical inference deals with data that have a fixed probability based on the number of cases and events. Bayesian inference deals with data whose probability is not fixed. That is, the probability is subject to change based on other factors. In this post, we will only discuss classical inference.

Classical Inference

If I roll a fair D20, then we know that the likelihood of getting a natural 20 is 0.05, or 5%. This means that if I roll the die 20 times, I should only get a natural 20 once. In fact, the probability of rolling each number is 5% and the P(<20) = 0.95 or 95%.

Let’s say I roll the die 20 times. If I get a natural 20 once, that is luck. If I get it twice then the chances of that happening naturally is P(20) x P(20) = 0.05 x 0.05 = 0.0025. The chances of that happening are smaller, so it sticks out more from normal.

Classical inference tells us that getting two natural 20s is statistically less likely than just one 20. Inference tells us this is a significant event. This is an example of classical inference because the probability is fixed (0.05) for each roll. Getting more than one natural 20 out of 20 rolls is statistically different from the expected, or model, probability.

Now, the probabilities for continuous variables, such as the probability of body temperature affecting the roll of the die, requires more advanced topics such sampling distributions and linear regression models.

You can read more about inferential statistics here.

The Takeaways

At its most basic, the probability is how likely an event is to occur. You can calculate the probability of an event with enough information about numbers and the event itself. The application of probability to collected data is statistics. Statistics makes sense and models of data for decisions about the event. I hope to see your questions below. Happy statistics!

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