4.0: Intro to Unit 4: Probability, Random Variables and Probability Distributions
Image from Funny Junk
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 🤗.
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.
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
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.
Image from Statistics How To
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.
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.
Two events are said to be independent if the outcome of one does not affect the outcome of the other. If two events are affecting each other, it makes the probability of calculating their likelihood a lot more complicated, so knowing that two events are independent usually makes our work more simple and effective.
For example, if I flip two coins, the likelihood of one landing on heads is not affected by the other coin. Therefore, we would say that these two events are independent. On the flip side, let’s consider temperature and snow likelihood. If the temperature is extremely low, the probability of it snowing will increase. Therefore, these two events are not independent, or dependent, since the temperature does affect the likelihood of snow.
Another key concept in probability is when two events are mutually exclusive. When two events are mutually exclusive, it means that it is impossible for them to occur at the same time.
To stay with our weather examples, the likelihood of having a hot day and snowing is impossible. Therefore, those two events are mutually exclusive.
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! 😊
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 distributions are events that involve four conditions:
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.
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.
For example, if we flip a coin until we get a heads this would represent a geometric distribution.