Introduction to Probability

[Photo by Ella’s Dad.]

Throughout history, around the world, people in every culture have enjoyed playing games of chance. Strangely, mathematicians did not begin to study chance and probability until the 17th century.

What Is Probability?

Probability is hard to define. Sometimes probability measures how strongly a person believes a certain statement. In other cases, probability measures how likely it is that a certain event will happen, if the assumptions we make in calculating the probability are true — if the dice are fair, the cards are fully shuffled, the draw is truly random, etc.

Probability doesn’t predict the future: incredibly improbable events happen all the time. You could say that probability is a measure of uncertainty, since if we knew what was going to happen, we wouldn’t need to estimate whether an outcome is likely or not.

How to Calculate Probability

Probability is usually given as a fraction between zero (an impossible event) and one (an event that is absolutely certain to happen) or as a percent between 0% and 100%. Remember, percents are a special type of fraction.

Once you learn how to count, it’s easy to calculate simple probability, IF all the events are all equally likely:

Probability \left( event A \right) = \frac{how \: many \: ways \: A \: can \: happen}{total \: number \: of \: possible \: events}

But if the events are not all equally probable, or if you want to know how likely it is that two or more events happen together, or that one event happens given another event, things get tricky. Are the events independent of each other, or does one depend on the other? Are they mutually exclusive? So many things to consider!

I made a tip sheet to help my MathCounts students remember the basics:

Probability Puzzles for You

Here is a simple probability puzzle from the July/August 1999 issue of my math newsletter:

In the game of Monopoly, what is the chance of taking a ride on the Reading Railroad on your very first turn?

And here are several harder questions about counting and probability:

Educate Yourself

Learn more about counting and probability with these challenging workbooks from AGMath:

Or explore the Dr. Math archives:

If you prefer video learning, try the excellent series of AoPS Counting & Probability Videos.

Or check out the Khan Academy probability playlist:

To Be Continued…

Read all the posts from the July/August 1999 issue of my Mathematical Adventures of Alexandria Jones newsletter.

10 thoughts on “Introduction to Probability

  1. Thanks for the reminder! I tend to take that for granted and forget to say it, but of course the students won’t realize that’s part of the definition if we don’t point it out.

    I suppose I should fix the tip sheet to make that clear as well, but I’m too busy today…

  2. Hi, old friend. I hope you do not mind this group message for the Carnival of Homeschooling:

    This post is in the 239th Carnival of Homeschooling, history of home education edition, which now is up at The Common Room, My theme is ‘the history of homeschooling in America.’ It was very interesting to research and I learned some fascinating things along the way (do you know why we have age segregated classrooms in America?).

    Please pay us a visit, and reciprocate in the publicity the carnival brings you by passing along the link along so others can visit as well.

    Please consider other ways to spread the news about the carnival as well- the more people who visit the carnival, the more link-love you get- If you have a facebook account, you could pass on the link there (fb doesn’t like tiny url links, so here’s the long one:, and if you have a twitter account, please tweet!


  3. Thank you for including my post in the Carnival! I’ve enjoyed the CoH for years, and it was the very first entry in my Blog Parties for Teachers sidebar widget. In fact, it was the reason I started the widget, so as to have a link always handy and stop cluttering up my blog posts.

  4. I like the videos, although I did miss the fact that there were three of them on my first reading. To everyone else: If you go to April’s blog and watch the Thinkwell sample, each video is pretty short. When you get to the end of one video, click the Forward button to jump to the next one.

  5. Great video! With my students I rather use the terms “number of favorable event” versus “total number of possible events”, that way has worked great for me to get the idea to stick in their minds.

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