Sample Spaces and Theoretical Probabilities

1. SECTION 4-3
Sample Spaces and Theoretical Probability

2. ESSENTIAL QUESTIONS
How do you determine sample spaces using various
methods?
How do you find theoretical probabilities?
Where you’ll see this:
Sports, food service, games, number theory

3. VOCABULARY
1. Event:
2. Sample Space:
3. Tree Diagram:
4. Fundamental Counting Principle:
5. Theoretical Probability:

4. VOCABULARY
1. Event: Any possible outcome or combination of possible
outcomes of an experiment
2. Sample Space:
3. Tree Diagram:
4. Fundamental Counting Principle:
5. Theoretical Probability:

5. VOCABULARY
1. Event: Any possible outcome or combination of possible
outcomes of an experiment
2. Sample Space: The set of all possible outcomes of the
experiment
3. Tree Diagram:
4. Fundamental Counting Principle:
5. Theoretical Probability:

6. VOCABULARY
1. Event: Any possible outcome or combination of possible
outcomes of an experiment
2. Sample Space: The set of all possible outcomes of the
experiment
3. Tree Diagram: A diagram of collections of branches used to
determine sample space
4. Fundamental Counting Principle:
5. Theoretical Probability:

7. VOCABULARY
1. Event: Any possible outcome or combination of possible
outcomes of an experiment
2. Sample Space: The set of all possible outcomes of the
experiment
3. Tree Diagram: A diagram of collections of branches used to
determine sample space
4. Fundamental Counting Principle: If there are two or more
stages of an activity, the total number of possible
outcomes is the product of possible outcomes of each
stage
5. Theoretical Probability:

8. VOCABULARY
1. Event: Any possible outcome or combination of possible
outcomes of an experiment
2. Sample Space: The set of all possible outcomes of the
experiment
3. Tree Diagram: A diagram of collections of branches used to
determine sample space
4. Fundamental Counting Principle: If there are two or more
stages of an activity, the total number of possible
outcomes is the product of possible outcomes of each
stage
5. Theoretical Probability: The way something should turn out

9. THEORETICAL VS.
EXPERIMENTAL

10. THEORETICAL VS.
EXPERIMENTAL
Theoretical Probability:

11. THEORETICAL VS.
EXPERIMENTAL
Theoretical Probability: The way things should happen

12. THEORETICAL VS.
EXPERIMENTAL
Theoretical Probability: The way things should happen
Experimental Probability:

13. THEORETICAL VS.
EXPERIMENTAL
Theoretical Probability: The way things should happen
Experimental Probability: The way things actua!y happen

14. EXAMPLE 1
Matt Mitarnowski is making a sandwich, using a bread
and a filling. He has three types of bread to choose from:
wheat (W), rye (R), or pumpernickel (P). He also has six
kinds of fillings: tuna (T), egg (E), shrimp (S), chicken (C),
ham (H), or bologna (B). How many possible sandwich
combinations are there?

15. EXAMPLE 1
Matt Mitarnowski is making a sandwich, using a bread
and a filling. He has three types of bread to choose from:
wheat (W), rye (R), or pumpernickel (P). He also has six
kinds of fillings: tuna (T), egg (E), shrimp (S), chicken (C),
ham (H), or bologna (B). How many possible sandwich
combinations are there?
FCP:

16. EXAMPLE 1
Matt Mitarnowski is making a sandwich, using a bread
and a filling. He has three types of bread to choose from:
wheat (W), rye (R), or pumpernickel (P). He also has six
kinds of fillings: tuna (T), egg (E), shrimp (S), chicken (C),
ham (H), or bologna (B). How many possible sandwich
combinations are there?
FCP: (3)

17. EXAMPLE 1
Matt Mitarnowski is making a sandwich, using a bread
and a filling. He has three types of bread to choose from:
wheat (W), rye (R), or pumpernickel (P). He also has six
kinds of fillings: tuna (T), egg (E), shrimp (S), chicken (C),
ham (H), or bologna (B). How many possible sandwich
combinations are there?
FCP: (3)(6)

18. EXAMPLE 1
Matt Mitarnowski is making a sandwich, using a bread
and a filling. He has three types of bread to choose from:
wheat (W), rye (R), or pumpernickel (P). He also has six
kinds of fillings: tuna (T), egg (E), shrimp (S), chicken (C),
ham (H), or bologna (B). How many possible sandwich
combinations are there?
FCP: (3)(6) = 18 combinations

19. EXAMPLE 1
Matt Mitarnowski is making a sandwich, using a bread
and a filling. He has three types of bread to choose from:
wheat (W), rye (R), or pumpernickel (P). He also has six
kinds of fillings: tuna (T), egg (E), shrimp (S), chicken (C),
ham (H), or bologna (B). How many possible sandwich
combinations are there?
FCP: (3)(6) = 18 combinations
Pairs:

20. EXAMPLE 1
Matt Mitarnowski is making a sandwich, using a bread
and a filling. He has three types of bread to choose from:
wheat (W), rye (R), or pumpernickel (P). He also has six
kinds of fillings: tuna (T), egg (E), shrimp (S), chicken (C),
ham (H), or bologna (B). How many possible sandwich
combinations are there?
FCP: (3)(6) = 18 combinations
Pairs: (W, T), (W, E), (W, S), (W, C), (W, H), (W, B), (R, T),
(R, E), (R, S), (R, C), (R, H), (R, B), (P, T), (P, E), (P, S),
(P, C), (P, H), (P, B)

21. EXAMPLE 2
Three coins (a penny, a nickel, and a dime) are tossed at
the same time. What is the probability of getting a heads
on exactly two of the coins?

22. EXAMPLE 2
Three coins (a penny, a nickel, and a dime) are tossed at
the same time. What is the probability of getting a heads
on exactly two of the coins?
Penny

23. EXAMPLE 2
Three coins (a penny, a nickel, and a dime) are tossed at
the same time. What is the probability of getting a heads
on exactly two of the coins?
Penny Nickel

24. EXAMPLE 2
Three coins (a penny, a nickel, and a dime) are tossed at
the same time. What is the probability of getting a heads
on exactly two of the coins?
Penny Nickel Dime

25. EXAMPLE 2
Three coins (a penny, a nickel, and a dime) are tossed at
the same time. What is the probability of getting a heads
on exactly two of the coins?
Penny Nickel Dime
H
T

26. EXAMPLE 2
Three coins (a penny, a nickel, and a dime) are tossed at
the same time. What is the probability of getting a heads
on exactly two of the coins?
Penny Nickel Dime
H
T

27. EXAMPLE 2
Three coins (a penny, a nickel, and a dime) are tossed at
the same time. What is the probability of getting a heads
on exactly two of the coins?
Penny Nickel Dime
H
H
T
H
T
T

28. EXAMPLE 2
Three coins (a penny, a nickel, and a dime) are tossed at
the same time. What is the probability of getting a heads
on exactly two of the coins?
Penny Nickel Dime
H
H
T
H
T
T

29. EXAMPLE 2
Three coins (a penny, a nickel, and a dime) are tossed at
the same time. What is the probability of getting a heads
on exactly two of the coins?
Penny Nickel Dime
H
T
H
H
H
T T
H H
T
T
T
H
T

30. EXAMPLE 2
Three coins (a penny, a nickel, and a dime) are tossed at
the same time. What is the probability of getting a heads
on exactly two of the coins?
Penny Nickel Dime
H HHH
T HHT
H
H HTH
H
T T HTT
H H THH
T
T THT
T
H TTH
T TTT

31. EXAMPLE 2
Three coins (a penny, a nickel, and a dime) are tossed at
the same time. What is the probability of getting a heads
on exactly two of the coins?
Penny Nickel Dime
H HHH
T HHT
H
H HTH
H
T T HTT
H H THH
T
T THT
T
H TTH
T TTT

32. EXAMPLE 2
Three coins (a penny, a nickel, and a dime) are tossed at
the same time. What is the probability of getting a heads
on exactly two of the coins?
Penny Nickel Dime
H HHH
T HHT
H
H HTH P(2 H) =
H
T T HTT
H H THH
T
T THT
T
H TTH
T TTT

33. EXAMPLE 2
Three coins (a penny, a nickel, and a dime) are tossed at
the same time. What is the probability of getting a heads
on exactly two of the coins?
Penny Nickel Dime
H HHH
T HHT
H 3
H HTH P(2 H) = 8
H
T T HTT
H H THH
T
T THT
T
H TTH
T TTT

34. THEORETICAL PROBABILITY

35. THEORETICAL PROBABILITY
Number of favorable outcomes
P( E ) =
Number of possible outcomes

36. EXAMPLE 3
Fuzzy Jeff has 3 shirts (1 blue, 1 black, and 1 white) and 2
pairs of jeans (one blue and one black), which he likes to
mix and match. If Jeff ’s outfit today is one of these shirts
and one pair of these jeans, what is the probability that
his shirt and jeans will be the same color?

37. EXAMPLE 3
Fuzzy Jeff has 3 shirts (1 blue, 1 black, and 1 white) and 2
pairs of jeans (one blue and one black), which he likes to
mix and match. If Jeff ’s outfit today is one of these shirts
and one pair of these jeans, what is the probability that
his shirt and jeans will be the same color?
(Blue, Blue), (Blue, Black), (Black Blue), (Black, Black),
(White, Blue), (White, Black)

38. EXAMPLE 3
Fuzzy Jeff has 3 shirts (1 blue, 1 black, and 1 white) and 2
pairs of jeans (one blue and one black), which he likes to
mix and match. If Jeff ’s outfit today is one of these shirts
and one pair of these jeans, what is the probability that
his shirt and jeans will be the same color?
(Blue, Blue), (Blue, Black), (Black Blue), (Black, Black),
(White, Blue), (White, Black)

39. EXAMPLE 3
Fuzzy Jeff has 3 shirts (1 blue, 1 black, and 1 white) and 2
pairs of jeans (one blue and one black), which he likes to
mix and match. If Jeff ’s outfit today is one of these shirts
and one pair of these jeans, what is the probability that
his shirt and jeans will be the same color?
(Blue, Blue), (Blue, Black), (Black Blue), (Black, Black),
(White, Blue), (White, Black)

40. EXAMPLE 3
Fuzzy Jeff has 3 shirts (1 blue, 1 black, and 1 white) and 2
pairs of jeans (one blue and one black), which he likes to
mix and match. If Jeff ’s outfit today is one of these shirts
and one pair of these jeans, what is the probability that
his shirt and jeans will be the same color?
(Blue, Blue), (Blue, Black), (Black Blue), (Black, Black),
(White, Blue), (White, Black)
P(Both same color ) =

41. EXAMPLE 3
Fuzzy Jeff has 3 shirts (1 blue, 1 black, and 1 white) and 2
pairs of jeans (one blue and one black), which he likes to
mix and match. If Jeff ’s outfit today is one of these shirts
and one pair of these jeans, what is the probability that
his shirt and jeans will be the same color?
(Blue, Blue), (Blue, Black), (Black Blue), (Black, Black),
(White, Blue), (White, Black)
2
P(Both same color ) =
6

42. EXAMPLE 3
Fuzzy Jeff has 3 shirts (1 blue, 1 black, and 1 white) and 2
pairs of jeans (one blue and one black), which he likes to
mix and match. If Jeff ’s outfit today is one of these shirts
and one pair of these jeans, what is the probability that
his shirt and jeans will be the same color?
(Blue, Blue), (Blue, Black), (Black Blue), (Black, Black),
(White, Blue), (White, Black)
2 1
P(Both same color ) = =
6 3

43. HOMEWORK

44. HOMEWORK
p. 160 #1-22
“Sometimes the questions are complicated and the
answers are simple.” - Dr. Seuss