Introduction to Probability

In this chapter you will be introduced to the world of counting (or combinatorics) and probability. Probability surrounds us in ways you might not expect. While casinos are an obvious example of probability in action, did you know that insurance companies use probability to determine your rates? Governments apply probabilistic methods when shaping policies on energy, environment, and budgeting. Even in your daily life, you unconsciously use probability when deciding whether to carry an umbrella or choosing the fastest route to work.

Historically the human race has been playing games with probability (and gambling) for over 5000 years (reference). The study of probability began earlier than Pascal and Fermat, but they are the ones that get the credit for bringing attention to the study through their own discussions in the mid 1600’s. Their work (with countless others) is what allowed for not only the study of counting and probability, but also gave birth to the field of statistics.

Today, probability theory has applications far beyond its origins in games and gambling:

• Weather forecasting: Meteorologists use probability models to predict tomorrow’s weather.
• Medical research: Doctors rely on probability to determine the effectiveness of new treatments.
• Financial planning: Investors use probability to assess risks and potential returns.
• Technology: From spam filters in your email to recommendation systems on streaming platforms, probability algorithms are at work.

Just for fun before you get started – the Monty Hall Problem is a rather famous for stumping a good number of mathematicians when it appeared in an article in the American Statistician in 1975 and later in Parade magazine in the 1990s. It is based upon a game played on the game show Let’s Make a Deal (with host Monty Hall). Read about the history of this problem at Wikipedia. Here is the description of the Monty Hall Problem from Marilyn vos Savant’s “Ask Marilyn” column in Parade magazine:

What do you think? Does it matter on the choice? Watch the video below to reveal the correct choice.

Numb3rs Math for Beginners (2 mins 07 secs – CC).

Learning Objectives

Below are the learning objectives for each section of the chapter.

2.1: Counting Methods

• Solve counting problems using the Addition Principle.
• Solve counting problems using the Multiplication Principle.
• Solve counting problems using permutations involving n distinct objects.
• Solve counting problems using combinations.

2.2 Probability Fundamentals

• Define a sample space for a random experiment.
• Identify events for a random experiment.
• Compute theoretical and experimental probabilities for events.
• Compute probabilities for events with equal likely outcomes.

2.3 Working with Events

• Solve probability applications using the complement rule.
• Find the probability when there is a union of two events.
• Find the probability when there is an intersection of two events.

2.4 Conditional Probability

• Compute conditional probabilities
• Compute probability using the multiplication rule
• Compute probability with independent events

2.5 Expected Value

• Compute expected values

Attributions

• This page contains modified content from David Lippman, “Math In Society, 2nd Edition.” Licensed under CC BY-SA 4.0.
• This page contains content by Robert Foth, Math Faculty, Pima Community College, 2021.