Empirical Probability: Definition, Formula & Examples

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  • 0:00 What Is Probability?
  • 1:11 What Is Empirical Probability?
  • 2:07 Formula for Empirical…
  • 3:17 Examples of Empirical…
  • 5:02 Lesson Summary
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Lesson Transcript
Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Expert Contributor
Michael Lawlor

Mike has 13 years of teaching experience at the college level. He has a Ph.D. in Statistics from Purdue University. Probability and modeling are his favorite areas.

In this lesson, we're going to explore empirical probability. We'll become familiar with the definition, look at some examples, and use the formula for empirical probability to become comfortable calculating these types of probabilities.

What Is Probability?

If you've ever wondered how likely it is that a certain event would occur, then you've wondered about probabilities. The probability of an event is the likelihood that the event will happen. Probability can be expressed as a fraction, decimal, or percentage. In this lesson, we will concentrate on probability in fraction or decimal form.

In decimal form, the range of possible probabilities is from 0 to 1. If an event has a probability of 0, then there is no chance of that event happening. If an event has a probability of 1, then the event is guaranteed to happen. The closer the probability of an event is to 1, the more likely the event is to happen, and similarly, the closer the probability of an event is to 0, the less likely it is to happen. For example, an event having probability 0.78 is more likely to happen than an event having probability 0.14, because 0.78 is closer to 1 than 0.14.

What Is Empirical Probability?

Let's concentrate on a specific type of probability called empirical probability. The empirical probability of an event is found through observations and experiments. It is the likelihood that the event will happen based on the results of data collected. For example, suppose we were to survey a group of 50 chefs. We ask them to choose their favorite type of food, such as the Italian, Mexican, French, American, or Chinese choices.

If 14 of the chefs chose Chinese as their favorite food, then based on these results, the empirical probability that a chef in this group would prefer Chinese food over all other types of cuisine would be 0.28. This is an example of empirical probability because we carried out an actual observation by collecting data through our survey. Our probability is based on the results of this data.

Formula for Empirical Probability

While we were discussing the food ballot, you may have wondered how the probability of a chef choosing Chinese food as their favorite type of cuisine was found. In empirical probability, we have a formula we use to calculate probabilities: we calculate empirical probability by dividing the number of times an event occurred during our experiment or observation by the total number of trials or observations.

Looking back at our food example, we found that 14 of the chefs chose Chinese food as their favorite type of cuisine. This is the number of times that the event of Chinese food being chosen as a favorite occurred. We also know that we asked 50 chefs to choose their favorite type of food. This is the total number of observations. Therefore, our empirical probability formula shows that the probability of a chef in this group choosing Chinese food as their favorite food is 14 / 50. To convert our formula into decimal form, we simply carry out the division: 14 divided by 50 equals 0.28. The results tell us that the probability of us choosing a chef out of this group whose favorite food is Chinese is 0.28.

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Additional Activities

Experiment 1:

Grab a coin. Have a student flip it 1 time.

A) What is the empirical probability of heads?

B) What is the empirical probability of tails?

C) Do these make sense logically?

The more you flip the coin, the closer the empirical probability will be to the true probability.

Repeat A and B above, but using n = 20.

If you have done hypothesis testing, you can test to see whether or not this coin is fair (unbiased). If you have not, do you believe whatever you got in part A is close enough to .5 to believe the coin is unbiased?

Experiment 2:

Put slips of paper with the numbers 1 - 6 in a hat and shake them up. Vary the frequency of each of these numbers, but have at least 30 pieces of paper total. This will simulate a (possibly) biased die.

Have a student draw 10 numbers from this hat with replacement and have them calculate the empirical probabilities of 1-6.

Have a student draw 10 numbers from this hat without replacement and calculate the empirical probabilities again.

Ask them which of these 2 do they believe is more accurate.

Experiment 3:

Ask your students what their favorite kind of pet is. Then calculate the empirical probabilities of dog, cat, rabbit, et cetera.

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