Unit 4 Overview: Probability, Random Variables, and Probability Distributions

 

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"Probabilistic reasoning allows statisticians to quantify the likelihood of random events over the long run and to make statistical inferences. Simulations and concrete examples can help students to understand the abstract definitions and calculations of probability. This unit builds on understandings of simulated or empirical data distributions and fundamental principles of probability to represent, interpret, and calculate parameters for theoretical probability distributions for discrete random variables. Interpretations of probabilities and parameters associated with a probability distribution should use appropriate units and relate to the context of the situation." -- College Board

As highlighted by the College Board's blurb of Unit 4, probabilistic reasoning is an important aspect of statistical analysis, as it allows statisticians to make predictions about the likelihood of certain events occurring based on data and probability theory. Simulations and concrete examples can be useful for helping students to understand and apply the abstract concepts of probability and probability distributions. 🧠

In this unit, students learn about theoretical probability distributions, which are used to describe the possible values and likelihoods of a random variable. These distributions can be either discrete or continuous, and they are characterized by specific parameters that describe the shape and behavior of the distribution.

It's important for you and me to understand the context and units of the probabilities and parameters associated with a probability distribution, as this can help them to interpret and apply their analyses to real-world situations!

Probability: What are the Odds?

Have you ever wondered how meteorologists determine the 🌧️ or ❄️ forecasts? What about the likelihood of a sports team winning a game? Analysts like meteorologists or sports analysts use probability models based on similar conditions in the past to predict the likelihood of these things happening in the present!  In this unit, you will learn some basics of probability and get a taste of what these statisticians use everyday to keep us safe and sound. 🤗

Probability is the study of possible outcomes and determining the chance of something happening. It is an essential part of statistics since we use probability as one of the main factors in making predictions or testing claims, which is what statistics is all about.

Categorical Variables

The most common type of probability you will encounter in this unit will deal with categorical variables. Recall from Unit 1 and Unit 2 that categorical variables are often represented with frequency tables or two-way tables (example pictured below). There are some important rules for determining probabilities from these types of displays that are essential to know in order to be successful on the AP exam. 🪑

Image from Statology

Quantitative Variables

The other type of variable that you will encounter is quantitative variables. Quantitative variables will generally be dealt with using density curves (example pictured below), most notably the normal distribution. The normal distribution is the most useful tool in statistics and hinges on a good understanding of probability. 🔔

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Probability Rules

There are many important rules and conditions that come into play when determining the probability of certain events happening. In order to be successful on the AP Exam, it is important to familiarize yourself with these rules and conditions.

Independence

The most important probability condition that you need to be aware of is the concept of independence. This will also be essential as we progress to inferential statistics in Units 6-9. 

In order to determine whether two events are independent, it's important to consider whether the outcome of one event could potentially affect the outcome of the other. If the outcome of one event has no effect on the outcome of the other, then the events are independent. On the other hand, if the outcome of one event could potentially affect the outcome of the other, then the events are dependent.

For example, in the case of flipping two coins, the outcome of one coin flip has no effect on the outcome of the other, so these events are independent. However, if we consider the probability of it raining on a given day, this probability may be affected by the temperature and other weather conditions. In this case, the events of it raining and the temperature are dependent (or not independent), as the temperature can affect the likelihood of it raining. 🌧️

Mutually Exclusive

Another key concept in probability is when two events are mutually exclusive. Mutually exclusive events are events that cannot occur simultaneously. In other words, if one event occurs, it is not possible for the other event to occur at the same time.

An example of mutually exclusive events is the outcome of rolling a single die. If the die is rolled, the outcome can only be one of the six possible numbers, so the events of rolling a 1, 2, 3, 4, 5, or 6 are all mutually exclusive. If the die is rolled and the outcome is a 4, it is not possible for the outcome to be any of the other numbers at the same time. 🎲

Another example: the likelihood of having a hot day and snowing is impossible. Therefore, those two events are mutually exclusive!

Probability Distributions

There are three types of probability distributions we will mainly focus on in this unit:  normal distributions, binomial distributions and geometric distributions. All of these have handy calculator functions that will make our work SO much easier! 😊

Normal Distribution

The most popular type of distribution in all data situations is the normal distribution. Whether it be ACT scores, heights of people or blood pressure levels, these all follow normal distributions and make it much easier to calculate where one data point compares to the rest of our data.

Binomial Distribution

It's time to meet a new character in the ever-growing tale of statistics! The binomial distribution is a probability distribution that is used to model the outcome of a series of independent, binary (two-outcome) events. It is characterized by four conditions: 2️⃣

1. Two possible outcomes (binary): The events being modeled must have only two possible outcomes, such as "success" and "failure" or "heads" and "tails."
2. Independent trials: The outcome of each event must not be affected by the outcome of any other event in the series.
3. Fixed number of trials: The total number of events in the series must be fixed and known in advance.
4. All trials are equally likely of occurring: The probability of each event occurring must be the same for all events in the series. The binomial distribution is used to determine the probability of a certain number of successes occurring within a fixed number of trials. For example, if you wanted to know the probability of flipping a coin 12 times and getting 10 heads, you could use a binomial distribution to model this.

Binomial distributions are events that involve four conditions:

• Two possible outcomes (binary)
• Independent trials
• Fixed number of trials
• All trials are equally likely of occurring Binomial distributions come in handy when you want to determine the likelihood of a certain number of successes within our fixed number of trials.

For instance, if you wanted to determine the likelihood of flipping a coin 12 times and receiving 10 heads, a binomial distribution would be appropriate.

Geometric Distribution

A geometric distribution is very similar to a binomial distribution, with the only difference being that we do not have a fixed number of trials. A geometric distribution typically involves repeating an action until you get a success. In other words, a geometric distribution models an indefinite number of trials until a success is achieved. 💎

An example of a situation that could be modeled by a geometric distribution is flipping a coin until you get a heads. The geometric distribution would be used to model the number of coin flips needed to achieve a heads.

🎥 Watch: AP Stats Unit 4