What is Probability Theory?

Probability theory can be summed up in one sentence: it is the part of mathematics that deals with probability. The picture below shows some examples of the uses of probability within multiple disciplines.
probability_theory
As you can see, probability theory touches upon many different disciplines. In other words, probability has a broad range of applications. Because probability is so broad, it also has different degrees of difficulty. Take a look at the table below to see which difficulty level you’re interested in learning. 
 
Concepts Difficulty Level
Simple probabilities of an event 1
Conditional probability & Bayes’ Theorem 2
Sensitivity, specificity, likelihood ratio 3
Probability using distributions 4

 

This section will cover some elements in all of these levels of difficulty.

 

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Random Variable

Random variables are the basic building block in probability theory. Random variables are distinct and should not be confused with traditional variables. The image below shows illustrates some of the major differences between the two.

random_variable_definition

As you can see, traditional variables are quite different from random variables. The table below explains these differences.

 

Type Definition Notation Example
Traditional Variable A symbol for an unknown value Lower case letters Height, density
Random Variable A variable whose outcome is unknown Upper case letters Roll of a dice, toss of a coin

 

While traditional values are typically measured or calculated, random variables are unknown until either an experiment is run or through the use of the probability.

 

Expected Value

Let’s say that a factory produces rain jackets and is interested in understanding whether or not the production of these jackets are uniform. The first step the factory takes is to measure the length of each rain jacket, giving them the following.

probability

This sample of 18 jackets gives them an idea of the average size of the jackets they produce. In other words, this sample mean tells them what jacket length they can expect from the factory.

 

Similarly, the expected value of a random variable can be thought of as similar to the mean of a traditional variable. Take a deck of cards as an example.

 

Card Value Number Included in Deck
1 10
2 5
3 3
4 2

 

The formula for expected value can be seen below.

expected_value

The elements in the formula are broken down in the table below.

 

Element Description
E[X] Expected value of random variable X
\mu Mean
\sum x P (X=x) Sum of all x’s times their probabilities, where X is taken to equal x

 

In this case, the expected value for the deck of cards would be:

 

Card Value Number Included in Deck Probability x * P
1 10 \frac{10}{20} 0.5
2 5 \frac{5}{20} 0.5
3 3 \frac{3}{20} 0.45
4 2 \frac{2}{20} 0.4
Sum 20 1.85

 

In other words,

 

E[X] = 1* \frac{10}{20} + 2* \frac{5}{20} + 3* \frac{3}{20} + 4* \frac{2}{20} =1.85

 

Probability Density Function

A probability density function is a function that lets us know the shape of the distribution. A probability distribution is a visualization that tells us what the probability of each value of a random variable is. Distributions can be uniform, exponential or normal.

 

Type PDF
Uniform \frac{1}{b+a}
Exponential \lambda e^{- \lambda x}
Normal \frac{1}{\sigma \sqrt{x \pi}} e^{\frac{1}{2}(\frac{x-\mu}{\sigma})^2}

 

exponential_probability_density
uniform_probability_density
normal_probability_density

 

Cumulative Distribution Function

A cumulative distribution function is a function that represents the area between two points under the curve on a continuous distribution. Recall that a probability distribution tells us the probability of a value occurring for each value a random variable can take on. The most common example of a probability distribution is a normal distribution.

standard_normal_example_3

The cumulative distribution function, or CDF, of the distribution above could tell us what the cumulative probability is between the two points, as shown by the highlighted area.

 

Types of Distributions

There are many different types of probability distributions. This is because different random variables have different values and different probabilities assigned to those values. Think about a coin toss, which only has two possible values: heads or tails. The probability of getting heads or tails is different as the number of trials increases. This distribution looks a lot different to the probability distribution of, say, the height of people in a population. The two probability distributions we’ll talk about here are listed in the table below, along with their formulas and examples.

 

Standard Normal N(\mu,\sigma) \frac{x-\mu}{\sigma} IQ scores in a population
Poisson Po(\lambda) \frac{\lambda^{x} e^{-\lambda}}{x!} Number of customers at a gas station per hour

poisson_distribution_example_1 distribution_example

 

Problem 1

You are interested in calculating the probability that someone has an IQ score of 110 given the following information.

 

Population average \mu 100
Population standard deviation \sigma 15

 

Solution 1

In order to calculate the probability of an IQ of 110, we need to first calculate the z-score, which is the formula in the table in the previous section.

 

    \[ z-score = \frac{110-100}{15} = 0.667 \]

 

In order to see what the probability of this z-score is on a standard normal curve, we have to look at a z-table. Since we’re interested in a left tail z-score - that is, all the probabilities up to 110 - we look at a left-tail z-table.

z-table_definition

This gives us a probability of 0.7454.

 

Problem 2

You need to create an annual report that gives people information on how many customers enter their store per day. In order to do this, you measure the following information.

  • The mean number of people per day is 15

 

Given this information, calculate the probability of 9 and 20 people entering the store on two separate days.

 

Solution 2

In order to calculate the probability of people entering the store, we simply plug in the mean, which is \lamda, into the equation for a Poisson distribution. Keep in mind Poisson distributions are those that deal with probabilities in time.

 

    \[ \frac{15^{9} e^{-15}}{9!} = 0.324 \]

 

For 200 customers, this probability is the following

 

    \[ \frac{15^{20} e^{-15}}{20!} = 0.0418 \]

poisson_distribution_example
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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.