Probability Models

In ancient times, oracles and soothsayers predicted the future based on tea leaves, animal behaviors, and other "signs." Today, we can predict the future using much more reliable probability models. Probability models have become so advanced today that they are used in the financial industry to predict changes in the economy. Some even claim that we could one day use machine learning and probability models to predict much more about the future. But probability models aren't just reserved for AIs -- we can use them too.

Probability models defined

A probability model shows us the mathematical chances of something happening in the future.

To build our probability model, we'll need the following elements:

A sample space: If we were flipping a coin, we would write the sample space for this very simple experiment as $\{H, T\}$ . One possible outcome is heads, and the other possible outcome is tails.
A set of all possible outcomes: We also need to make sure we're covering all possible outcomes. For example, there might be a chance that the coin lands on its edge. When written correctly, our sample space usually represents all possible outcomes.
A set of probabilities for each element in our sample space: We also need to know the chances of each possibility. For example, there is a 50% chance of getting either heads or tails when we flip a coin.

Sometimes, we don't even know the probabilities of each possibility -- but we might still have the opportunity to build a probability model.

Creating our own probability model

Now that we know exactly what we need, it's time to build our probability model.

Let's say that a smartwatch comes in three colors, and we can choose a matte finish or a glossy finish. As a result, there are six possible outcomes:

Matte green
Matte blue
Matte black
Glossy green
Glossy blue
Glossy black

But we're not finished building our probability model just yet. First, we need to figure out the likelihood of a customer choosing each option. Some might be more popular than others. For example:

25% of customers choose matte green
12.5% choose matte blue
37.5% choose matte black
5% choose glossy green
10% choose glossy blue
10% choose glossy black

Disjoints

If two things cannot happen at the same time, then they are called "disjoint." For example, it's not possible to choose a glossy blue smartwatch that is also matte green. These two possibilities are disjoint.

But it is possible to see two possible occurrences if we change the definition of our experiment. Let's say that our new experiment studies the purchases of people buying two watches simultaneously. Now we can say that one potential possibility is the purchase of a matte green and a glossy blue smartwatch. These possibilities are not disjoint. A simpler example involves flipping two coins at the same time.

Independent and dependent events

We might also need to change our probability model if we're dealing with independent events. An independent event is one that is not affected by any previous events. For example, the probability of getting heads when flipping a coin doesn't change based on the result of the previous flip. You always have a 50% chance of getting tails.

Interestingly, this also explains something called "gambler's fallacy." This is the belief that your chances of winning something based on independent events increase with each subsequent entry or roll. For example, some people believe that if they continue to enter the lottery every month, their chances of winning get higher and higher with each passing month. In truth, their chances remain exactly the same throughout the months because each lottery draw is an independent event.

But sometimes, the probability of an event does change based on previous events. For example, what would happen if there was only one type of each smartwatch in stock? The first customer to arrive might choose matte green. When the second customer arrives, there is now a 0% chance of them choosing the matte green smartwatch. The sample space would also change, which would also change the percentage associated with each option. If there are only 5 options, the probability of the customer choosing each watch is now 20%.

Topics related to the Probability Models

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Expected Value

Flashcards covering the Probability Models

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Practice tests covering the Probability Models

Probability Theory Practice Tests

Common Core: High School - Statistics and Probability Diagnostic Tests

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