Empirical vs. Theoretical Probability: Lesson Topic Index | Grade 7 Math | Intermediate Test Prep | StudyZone

 In this lesson we will examine two different types of probability, empirical and theoretical probability.   Although both of these methods for determining the probability of an event involve the concept of "chance", the solutions are obtained in two very different ways.

 Empirical Probability When you perform an experiment, or sample a number of people or objects to determine the probability of an event, you are determining the "empirical probability" of an event. For example, let's say that a manufacturer tested 1000 radios, at random, and found 15 of them to be defective. We can easily determine that the empirical probability that a radio is defective would be: P(defective radio) = 15                      1000 or...3/200.  As a decimal it would be .015, and as a percent it would be 1.5% Now the manufacturer can use this result to predict that in the production of 7500 radios, 1.5% of them will probably be defective. Or, (.015)(7500) = 112.5 defective radios. Let's look at another example of empirical probability. Let's say that Sheila tossed a coin 200 times and recorded whether it was a "head", or a "tail" after each toss. After the experiment, Sheila found that she had recorded 122 "heads" and 78 "tails." From this experiment she could predict that the empirical probability of getting a "head" is: P(Head) = 122            200 or..61/100. As a decimal it would be .61, and as a percent 61%. Because in determining empirical probability you actually perform an experiment, it is sometimes called: "experimental probability."

 Theoretical Probability An empirical approach is required whenever we want to determine the probability of an event by actually performing the experiment.  However, common sense tells us that there is also a simple, mathematical, way to determine the probability of an event in another way. For example, Jack is playing a game in which each player must roll a die.  To win the game Jack must roll a number greater than 4.  What is the probability that Jack will win on his next turn? Common sense tells us that: (a) The die has an equal chance of landing in 6 ways. (b) There are only 2 ways for Jack to win the game. (c) Therefore the probability of Jack winning is: P(Jack wins) = 2                6 or...1/3. As a decimal it would be .333..., as a percent, 33 1/3% And one more example of "theoretical probability." In this game a standard 52 card deck of playing cards is used.  In order to win you must pick a "face card." What is the probability that you will win the game on the next draw? Again, common sense tells us: (a) Each card in the deck has an equal chance of being drawn. (b) There are 12 face cards (winning cards) in the deck. Therefore the probability of winning on the next draw is: P(face card) = 12                 52 or... 3/13. As a decimal it would be about .23, and as a percent, 23%

An empirical example

 During the last 15 basketball games, Sam has made 64 and missed 32 foul shots. What is the empirical (or experimental) probability that Sam will make his next foul shot?

Using the statistical information in the problem we can determine that the empirical probability of Sam making his next foul shot would be:
P(making the next shot) = 64 (made)
96 (total)
or..2/3.
as a decimal it would be .666..., as a percent, 66 2/3%

A theoretical example

 In a box there are 4 red marbles, 3 blue marbles and 5 yellow marbles.  If one marble is chosen at random from the box, what is the probability of choosing: (a) a red marble (b) a blue or yellow marble

P(red) = 4
12
or 1/3, (decimal):.333.., (percent): 33 1/3%

P(blue or yellow) = 8
12
or 2/3, (decimal):.666.., (percent): 66 2/3%