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4 Tutorials that teach Probability Distribution
Common Core: S.MD.1

# Probability Distribution

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Author: Sophia Tutorial
##### Description:

This lesson will introduce probability distributions.

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Tutorial

In this tutorial, you're going to learn about probability distributions by focusing on:

1. Discrete Probability
2. Continuous Probability

## 1. DISCRETE PROBABILITY

This spinner has eight equally sized sectors. So if you spin the spinner, there's three sectors that say 1, one that says a 2, two of them that say 3, and another two that say 4. All the sectors are equally likely, but not every outcome is equally likely.

A probability distribution can be set up for the spinner. The probability distribution is a lot like a frequency distribution, except you will set it up as probabilities instead of frequencies. All the outcomes that could happen from the spinner will be listed...the 1, the 2, the 3, and the 4.

Next, instead of how often they come up in terms of frequency, we're going to list how often they come up in terms of probability. Three-eighths of the time you'll get a 1. One-eighth of the time you'll get a 2. Two-eighths of the time you'll get a 3. And two-eighths of the time you'll get a 4.

Two things should be noticed with this probability distribution:

Every probability-- this three-eighths, one-eighths, two-eighths, and two-eighths-- each are a number between 0 and 1. Any probability in a probability distribution has to be between 0 and 1 inclusive.

The sum of all the probabilities in the probability distribution is 1. It makes sense. A probability of 1, it's certain that you get a 1 or 2 or 3 or 4 on this spinner.

This type of probability is called discrete probability distributions, which means there only so many outcomes.

• Discrete Probability Distribution
• A probability distribution with only so many values. The probabilities can be listed in a table alongside the potential outcomes.
• Probability Distribution
• A description of the possible outcomes and their probability of occurring.

Consider two flips of a coin. Here are your four possibilities. Create a probability distribution for the number of tails.

What you should have come up with is this:

The probability that you get that is a fourth probability of 0, a half probability of 1-- because there were two of the four outcomes-- and one-fourth probability of two tails on the two flips. Notice once again, each of those probabilities is a number between 0 and 1. And one-fourth plus one-half plus one-fourth in fact does add to 1.

• 0 tails
• one tail
• two tails

The coin and spinner scenarios are called discrete probability distributions, which means there are only so many outcomes. One of the examples had four outcomes. One of them had three potential outcomes.

## 2. CONTINUOUS PROBABILITY

There are also probability distributions that are continuous probability distributions in which the probability is related by some mathematical function. Like the normal distribution is a probability distribution

• Continuous Probability Distribution
• A probability distribution where probabilities are related by a mathematical function, and the outcomes can take any value within a given range.

In this case, the area under the curve should equal one and the graph should lie entirely above or on the x-axis. That's how those two rules apply to the continuous distributions. With a continuous distribution, an outcome can be anything within this range on the x-axis, anything within the range of values here.

One last thing that's worth noting is that some distributions have what we call countably infinite outcomes.

You are interested in the number of rolls it takes to obtain a 6 on your die. If you rolled a six on your first roll that would be a 1, because it took you one roll. If you rolled it on your second trial after missing on the first trial, then it would be a 2.

But suppose we want 4, 2, 3, 2, 3, 4, 5, 2, and just kept on going here, hypothetically-- not really practically-- but hypothetically, this could go on forever. And so there are infinitely many outcomes here, or infinitely many rolls that it hypothetically could take to obtain a 6. And so this is what we would call countably infinite outcomes.

And these are considered discrete because, for instance, it can take you one and a half rolls to obtain a 6. It has to take an integer value. It has to take like a 1 or 2 or 3, it can't take a one and a half or 2.25 rolls. Since it can only contain these particular values as outcomes, the integers, it is discrete instead of continuous.

Probability distributions are a lot like frequency distributions in that they show the different outcomes. But instead of frequency distributions, we're not going to measure how often they take that value, but instead how likely each of those outcomes is. So there's going to be discrete probability distributions, and those are going to either have a finite or countably infinite number of outcomes. And we went through both of those scenarios in this tutorial.

Continuous probability distributions mean that outcomes can take any value within a given range. And so we talked about two types of probability distributions: discrete and continuous.

Good luck!

Source: This work is adapted from Sophia author jonathan osters.

Terms to Know
Continuous Probability Distribution

A probability distribution where probabilities are related by a mathematical function, and the outcomes can take any value within a given range.

Discrete Probability Distribution

A probability distribution with only so many values. The probabilities can be listed in a table alongside the potential outcomes.

Probability Distribution

A description of the possible outcomes and their probability of occurring.