1. Rolling Into Probability
Experimental vs. Theoretical Probability

2. Goal and Objectives
Goal: Students will roll two dice and to see the difference between experimental and theoretical probability.
Objective: Given two dice students will find different experimental probability values. Students will find the relationship between theoretical and experimental probability. Student’s will understand why they can differ.

3. Vocabulary
Probability- the chance that a particular outcome will occur, measured as a ratio of the total of possible outcomes
P = favorable outcomes
possible outcomes
Experimental Probability-  is the ratio of the number of times the event occurs to the total number of trials
Theoretical Probability- is the likeliness of an event happening based on all the possible outcomes
Sample Space-  is the set of all possible outcomes of that experiment

4. Materials
Dice
Charts
Worksheet
Pen/Pencil

5. Finding the Theoretical Probability
To first find your theoretical probability you need to know you sample space.
When rolling a dice we know that there are six sides, labeled by dots that are 1 thorough 6.
So if we want to find the probability of one dice we would know the sample space would be {1, 2, 3, 4, 5, 6}
What would be the probability of rolling a 3?
We would know it would be 1/6 probability

6. When using two dices what would you think to do
When using two dices what would you think to do? (don’t just add another six)
Rolling two fair dice more than doubles the difficulty of calculating probabilities. This is because rolling one die is independent of rolling a second one, which means one roll has no effect on the other one. When dealing with independent events we use the multiplication rule, therefore 6x6 = 36.
Our sample space is 36.
Example: Probability or rolling a 6 and a 6 is 1/36.
Important Note – rolling a 1,3 is different than rolling a 3,1.

7. Sample Space 1 2 3 4 5 6 (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)

8. Directions Everyone needs a partner, approximately six groups.
Partner one: The dice roller
Partner two: The result recorder
Partner one will roll two dice 36 times, 1 roll for each element in the sample space.
Partner two will record each roll in the chart provided.
Once all 36 rolls are complete the students will work together to fill in the probabilities that they have come up with for all the possible outcomes.
Look for similarities and differences in your chart and the theoretical chart provided.

9. Why do they differ?
Theoretical probability is based on formulas that intend to describe the probability of an event. Experimental probability is the based on the frequency of an event from some sample space.
These two figures can differ for the same event due to factors that aren't taken into consideration. Theoretical probability cannot exist without some uncertainty.
If one knows absolutely everything about an event, the result can be determined without any uncertainty. Chance is based on one's limited perspective.

10. One Last Question
Two fair six-sided dice are rolled. What is the probability that the sum of the two dice is seven?
Think of all the possibilities!

11. Solution
There are six ways to get a sum of seven with two dice: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1). There are a total of 36 outcomes, so the probability that we’re looking for is 6/36 = 1/6.

12. Connections to Real Life
Probability is used in Real Life not just Math Class we actually use it everyday.
Some examples
Predicting the Weather
Batting Averages in Baseball
Winning the Lottery

13. References