Experimental vs. Theoretical Probability: Flip a Coin

The comparison of experimental probability and theoretical probability was one of the topics we discussed in class; and for me, one of the easiest to grasp. I did find, however, that some of my classmates appeared to struggle with the concepts. So, this post will be my attempt to simplify the information so that anyone can distinguish between the two.

To begin, I offer basic definitions of some terms associated with this discussion (courtesy of my math professor):

• Theoretical Probability: the outcome under ideal conditions.
• Experimental (empirical) Probability: determined by observing outcomes of experiments.
• Sample Space: a set of all possible outcomes for an experiment.
• Equally Likely: when one outcome is as likely as the other.
• Mutually Exclusive Events: the events cannot be both at the same time.
• Complementary Events: two mutually exclusive events that total the sample space.
• Bernoulli's Theorem: if an experiment is repeated a large number of times, the experimental probability of a particular outcome approaches a fixed number as the number or repetitions increases (Law of Large Numbers.)

One simple way to demonstrate this, is by the use of something familiar to most of us - the flip of a coin. 

The theoretical probability of flipping a coin is 1/2. There are two possible outcomes when you flip a coin: heads or tails. The probability of the two outcomes together equals one; they are complementary events and are equally likely.

Image by me

The experimental probability is not as straight forward, however, much more fun to find! I am an extremely visual person, so I searched for simulations on the internet that would allow me to also show you experimental results of flipping a coin. Rather than type a book, I will provide visuals. There are several websites that offer virtual coin tosses: Random.org, Virtualcointoss.com, Shodor.org, and Justflipacoin.com to name a few. However, of these only Shodor.org actually tracks the ratios for you. I wanted more, so I went outside the box here - and I am so glad I did.

I found an adjustable spinner simulator on The National Council of Teachers of Mathematics (NCTM) website that is A-mazing. Here are just some of the reasons why:

1. adjustable number of sectors. This allowed me to change it to only two for the purpose of a coin-flip experiment.
2. adjustable name in the "color" column. This allowed me to assign Heads to blue, and Tails to yellow (the default was just the color name).
3. adjustable number of spins per spin. This expedites the time it takes to flip a coin many times.
4. "Skip to End" button. If you enter 100 spins per spin, this button will skip to the end of the results, instead of making you wait for the spinner to spin all 100 times.
5. BONUS: Experimental % and Theoretical % columns!!! How better a way to put it into perspective? This one feature captures the essence of why I am writing this post in the first place!

Now for the experiment. This first image is how I started the experiment. Note that the theoretical probability currently shows as a 50/50 percent ratio. As mentioned above, the outcome of heads or tails is equally likely.

Work by me

This next image is after 20 spins, or coin flips. Now you are able to see the amount of times I flipped a head versus a tail, as well as the experimental probability showing a 60/40 percent ratio. While this ratio is not very far off from the theoretical ratio of 50/50, watch what happens when we increase the number of spins.

Work by me

This last image is after 500 spins, or coin flips. Again, you are able to see the amount of times I flipped a head versus a tail, as well as the experimental probability showing a 49/51 percent ratio. By increasing the number of spins, my experimental probability became much more comparable to the theoretical probability. In other words, an example of Bernoulli's Theorem.

Work by me

I hope I simplified this topic for you! By making this an interactive math lesson, I believe you can also make it easy and fun for your students to understand. Using my example, the 20 spins could reflect the spins one of your students would make, and the 500 spins could be your class total. Make it fun!

-Amanda