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Probability Distribution
Common Core: S.MD.1

Probability Distribution

Author: Jonathan Osters

This lesson will introduce probability distributions.

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In this tutorial, you're going to learn about probability distributions. So let's take a look at this spinner. 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.

So we can set up a probability distribution for the spinner. Now probability distribution is a lot like a frequency distribution, except we're going to set it up as probabilities instead of frequencies. So we're going to list all the outcomes that could happen from the spinner, 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.

Now, what you should notice with this probability distribution is a couple of things. First, is that 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. Second, is that 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.

So let's put those rules into action on this next example. Consider two flips of a coin. Here are your four possibilities. Create a probability distribution for the number of tails. Pause the video and scribble it out.

What you should have come up with is this. You can get 0 tails, which is here. You can get one tails, which is here or here. Or you can get two tails right here. 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.

Now, the thing about the last two examples is that both of them were what we called discrete probability distributions, which means there are only so many outcomes. One of the examples have four outcomes. One of them had three potential outcomes. There are also though probability distributions that are continuous in which the probability is related by some mathematical function. Like the normal distribution is a probability distribution.

In this case, the area under the curve should equal one and the graph should lie entirely above or on the x-axis. And 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. For instance, suppose you were 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 we're going to consider discrete. We're going to consider them 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. And so because it only can contain these particular values as outcomes, the integers, we're calling it discreet instead of continuous.

And so to recap, 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.

We're also going to deal with continuous probability distributions in which the outcomes can take any value within a given range. And so we talked about probability distributions and they're two flavors, discrete and continuous. Good luck and we'll see you next time.

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.