Mathematical Models with Applications
HS Mathematics
Unit: 01 Lesson: 01
Empirical Probability
Empirical probability describes the relative frequency of actual observations of an event in an experiment. Empirical probability is also known as experimental probability. Empirical probability is mathematically described as:
( )
 number of times desired event occurs total number of trials or tested items
How does the mathematical definition of empirical probability compare with the mathematical definition of theoretical probability?
Empirical probability also follows the Laws of Probability.

A probability is a number between 0 and 1, inclusive.

The sum of all probabilities for a sample space of a situation is one. The probability of the remaining events in a sample space is called the complement and is found by
1 – P(E).

The probability of an event that must occur is one.

The probability of an event that cannot occur is zero.
In situations where theoretical probability cannot be calculated, empirical probability can be determined experimentally.
The accuracy of empirical probability depends on the number of trials. The Law of Large Numbers says that the more times an experiment is performed, the more accurately the probability can be used as a predictor.
©2012, TESCCC 05/15/12 page 1 of 4

Mathematical Models with Applications
HS Mathematics
Unit: 01 Lesson: 01
Empirical Probability
Guided Practice
MedcoMeds tested a new cholesterol medicine that was meant to lower levels of bad cholesterol.
The medication was given to 500 individuals with the following results:
Cholesterol lowered ----------------------- 260
Cholesterol increased -------------------- 82
Cholesterol stayed the same ----------- 158 a. Why is this considered empirical probability? b. Find P (lowered cholesterol levels). c. Find P (increased cholesterol levels). d. Find P (cholesterol levels stayed the same). e. How could the approximation of the probability be improved? Why would this be important in this situation?
©2012, TESCCC 05/15/12 page 2 of 4

Mathematical Models with Applications
HS Mathematics
Unit: 01 Lesson: 01
Empirical Probability
Independent Practice
1. What are the similarities in theoretical and empirical probabilities?
2. What are the differences in theoretical and empirical probabilities?
3. What is the Law of Large Numbers and how does it pertain to empirical probability?
4. Six friends chartered a fishing boat out of Port Aransas to go bay fishing. By noon they had caught 54 fish as shown in the chart.
Trout
Fish Number Caught
16
Redfish
Flounder
Shark
Croaker
Sheephead
14
4
2
12
6 a. What is the probability that the next fish caught will be a redfish? b. Is this an example of theoretical or empirical probability? Explain your reasoning.
5. A survey was taken of 6,845 married couples, both of which were 40 years of age. After 30 years 5,125 of the men were still alive and 5,600 of the women were still alive. a. What is the probability that a 40 year old married male will live to be 70 years of age? b. What is the probability that a 40 year old married female will live to be 70 years of age? c. What outside variables might affect the accuracy of this probability? Explain your reasoning.
©2012, TESCCC 05/15/12 page 3 of 4

Empirical Probability
Mathematical Models with Applications
HS Mathematics
Unit: 01 Lesson: 01
6. What is the theoretical probability of rolling an even number on a die? Explain how to find the empirical probability of rolling an even number on a die. Find the empirical probability for 40 trials. How does it compare to the theoretical probability?
7. Susan’s Sweets makes heart shaped cinnamon suckers for Valentine’s Day. In a sample of 120 suckers taken from the conveyor belt carrying the suckers, 8 were broken. What is the probability that if one sucker is taken off the conveyor belt, it will b e broken? If Susan’s
Sweets produces 825 heart shaped suckers, what would be the predicted number of broken suckers?
8. Spin a quarter on its edge. Note which side lands face up. Repeat for a total of 50 times.
What is the probability the quarter will land face up? Was the result different from expectations? What are some possible reasons for the results?
©2012, TESCCC 05/15/12 page 4 of 4