1. Probability & Sample Space

2. Probability
The probability of a number is a ratio that compares the number of successes to the total possible outcomes.

3. Probability
Probabilities can be expressed as a fraction, a decimal, or a percent.
0% % 100%
/2 1
Impossible Certain

4. Probability Something that will never happen has a 0% probability.
Exmaple: There is a 0% probability that humans will be able
to breath underwater (without an oxygen mask).
0% % 100%
/2 1
Impossible Certain

5. Probability Something that will always happen has a 100% probability.
Example: There is a 100% probability that water will freeze
At 32 degrees Fahrenheit.
0% % 100%
/2 1
Impossible Certain

6. Sample Space
In order to determine the probability of an event occurring, you must first determine the sample space.
The sample space of an event is all possible outcomes.

7. Example 1 Suppose your favorite color is teal.
If you were going to close your eyes and choose a balloon at random, what is the probability that you would choose your favorite color?

8. Example 1 First we must determine the sample space:
There are four possible outcomes: you could grab the peach one, the purple one, the pink one, or the teal one.

9. Example 1 The sample space is all possible outcomes: Peach Purple Pink
Teal

10. Example 1
The probability of grabbing the teal balloon is the number of teal balloons, 1, divided by the number of possible outcomes, 4.
P(teal) =

11. Example 1 P(peach) = P(purple) = P(pink) =
We can also find the probabilities for the other colors.
P(peach) =
P(purple) =
P(pink) =

12. Example 2
You are playing a game that involves one die. What is the probability of rolling a 6?
First we must determine the sample space.

13. Example 2 The sample space is all possible outcomes:
1, 2, 3, 4, 5, and 6.

14. Example 2
So the probability of rolling a 6 is:

15. Example 3
Suppose you were going to roll a tetrahedral die. What is the probability of rolling a 1?
Sample Space: 1, 2, 3, 4

16. Example 3
Since there are only four outcomes, the The probability of rolling a 1 is:

17. Example 4
Suppose you were going to toss two coins. What is the sample space?
H H
H T
T H
T T

18. Example 4 What is the probability of tossing two heads? H H H T T H

19. Example 4 What is the probability of tossing two tails? H H H T T H

20. Example 4 What is the probability of tossing a head and a tail? H H
H T
T H
T T

21. Example 5
You are playing a game that involves 2 dice. What is the sample space?

22. Example 5 On the first die, we can get a 1, 2, 3, 4, 5, or 6.
On the second die, we can get a 1, 2, 3, 4, 5, or 6.

23. Example 5
So the sample space is:

24. Example 5
Since there were six possible numbers we could roll for the first die, and six possible numbers we could roll for the second die,
there are a total of 6 x 6 = 36 possible combinations!

25. Example 6
You are going to buy a pizza. The pizzas come in three different sizes with three choices of seasoning.
Size
Seasoning
Small
Pepper
Medium
Basil
Large
Onions

26. Example 6 What is the sample space? Let’s use a counting tree to
help us determine all the
possible pizzas we
can create!

27. Example 6 So the sample space is: There were 3 choices for the
Small + Pepper
Small + Basil
Small + Onion
Medium + Pepper
Medium + Basil
Medium + Onion
Large + Pepper
Large + Basil
Large + Onion
There were 3 choices for the
size, and 3 choices for the
seasoning.
So there are a total of 3 x 3 = 9
possible combinations!

28. Example 7
A restaurant offers 5 main entrees, 7 side items, and 6 beverages.
How many different meals are possible?
Entree
Side
Beverage
Steak
Potatoes
Coffee
Chicken
Squash
Hot Tea
Shrimp
Beans
Iced Tea
Salmon
Carrots
Coca-Cola
Pork
Peas
Sprite
Okra
Milk
Spinach

29. Example 7 So there are 5 x 7 x 6 = 210 different possible meals.
For each meal, you must choose an entrée, a side, and a beverage.
So there are
5 x 7 x 6 = 210
different possible meals.
Entree
Side
Beverage
Steak
Potatoes
Coffee
Chicken
Squash
Hot Tea
Shrimp
Beans
Iced Tea
Salmon
Carrots
Coca-Cola
Pork
Peas
Sprite
Okra
Milk
Spinach