1. 5.2 Probability

2. Dice Later, we’re going to do some activities with partners
Go ahead and find a partner, and grab one die each
If you have a group of 3 people, still only grab 2 dice for the group

3. Probability Models
Last class, we used simulation to imitate chance behavior
Fortunately, we don’t have to rely on simulation to determine the probability of an event occurring
Instead, we have a method to calculate the probability directly

4. Sample Space
The sample space (S) of a chance process is the set of all possible outcomes
So what is the sample space for the process of flipping a fair coin one time?

5. Sample Space
The sample space (S) of a chance process is the set of all possible outcomes
So what is the sample space for the process of flipping a fair coin one time?
Heads and tails
Suppose you have a weighted coin that comes up heads 77% of the time. What is the sample space now?

6. Sample Space
The sample space (S) of a chance process is the set of all possible outcomes
So what is the sample space for the process of flipping a fair coin one time?
Heads and tails
Suppose you have a weighted coin that comes up heads 77% of the time. What is the sample space now?

7. Example You and your partner each have a die
Imagine that you each roll your die once
What are all the possible combinations that could result from this?

8. Example You and your partner each have a die
Imagine that you each roll your die once
What are all the possible combinations that could result from this?
36 combinations

9. Probability
What is the probability of getting a total of 5 on the 2 dice?
Let’s first simulate it: you and your partner should do 25 rolls
Track how many of them have a total of 5 on the dice
Then put them up on our table

10. Probability
What is the probability of getting a total of 5 on the 2 dice?
Let’s first simulate it: you and your partner should do 25 rolls
Track how many of them have a total of 5 on the dice
Then put them up in our table
Now, using the table, find the estimated probability that any particular roll of two dice will total 5

11. Probability We could have also done this without simulation
We know that the sample space (S) has 36 different possible outcomes
How many of them have a total of 5?

12. Probability 4 out of the 36 possible outcomes result in a sum of 5
Each outcome has the same probability of occurring
Assuming the dice are fair
So the probability of any of the 36 outcomes occurring individually is 1/36 or about .028
So what do you think is the probability of getting a sum of 5, if there are 4 different outcomes that would give us this result?

13. Probability 4 out of the 36 possible outcomes result in a sum of 5
Each outcome has the same probability of occurring
Assuming the dice are fair
So the probability of any of the 36 outcomes occurring individually is 1/36 or about .028
So what do you think is the probability of getting a sum of 5, if there are 4 different outcomes that would give us this result?
4/36 or about .11

15. Probability Models So a probability model does 2 things:
It defines the sample space (all possible outcomes)
It then provides a probability for each of those possible outcomes

17. Basic Rules of Probability
The probability of any event is between 0 and 1
Can be written as a fraction (1/36, 5/12, etc.)
Can be written as a decimal (.25, .01, etc.)
Usually not written as a percentage (.25 is the same as 25%, but we prefer to write is as .25)
All possible outcomes must have probabilities whose sum is 1
The probability that an event does not occur is 1 minus the probability that it does occur
If two events have no outcomes in common, the probability that ONE OR THE OTHER occurs is the sum of their individual probabilities
Example: the probability of rolling a 6 or a 2

18. An Example So, for our situation where we roll two dice:
A. what is the probability of rolling a (combined) 3?
B. What is the probability of NOT rolling a (combined) 3?
C. What is the probability of rolling (combined 3) OR a (combined) 2?

19. An Example So, for our situation where we roll two dice:
A. what is the probability of rolling a (combined) 3?
2/36
B. What is the probability of NOT rolling a (combined) 3?
34/36
C. What is the probability of rolling (combined 3) OR a (combined) 2?
3/36

20. Mutual Exclusivity
Two events are mutually exclusive if they have no outcomes in common and so can never occur together
On a single die, we can’t roll a 6 and a 4 at the same time
On a coin, we can’t flip heads and tails at the same time
At any given point during the day, you can’t be in math class AND in Spanish class
Etc.

27. Venn Diagram vs two-way table
They are telling you the same information, so you can calculate probabilities either way
I personally prefer the Venn Diagram, some people prefer the two-way table
If the question gives you one or the other, probably easiest to just go with that