1. Outline Definitions Approaches Axioms Sample-Point Approach
1. Sets and Probability
1.3 Probabilistic Model of an Experiment
1.4 Sample-Point Approach in Calculating Probability
Ruben A. Idoy, Jr.
Introduction to Probability Theory
(Math 181)
21 June 2012

2. Outline Definitions Approaches Axioms Sample-Point Approach
Outline
1 Definitions

3. Outline Definitions Approaches Axioms Sample-Point Approach
Outline
1 Definitions
2 Approaches of Probability Values

4. Outline Definitions Approaches Axioms Sample-Point Approach
Outline
1 Definitions
2 Approaches of Probability Values
3 Axioms of Probability

5. Outline Definitions Approaches Axioms Sample-Point Approach
Outline
1 Definitions
2 Approaches of Probability Values
3 Axioms of Probability
4 Sample-Point Approach on Calculating Probability
Steps
Examples

6. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
experiment - the process of making an observation.

7. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
experiment - the process of making an observation.
An experiment can result in one, and only one, of a set of distinctly
different observable outcomes.

8. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
experiment - the process of making an observation.
An experiment can result in one, and only one, of a set of distinctly
different observable outcomes.
We are interested in experiments that generate outcomes which vary in
random manner and cannot be predicted with certainty.

9. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
experiment - the process of making an observation.
sample space - denoted by S (or Ω in some books), is a set of points
corresponding to all distinctly different possible outcomes of an
experiment. Each point corresponds to a particular single outcome.

10. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
experiment - the process of making an observation.
sample space - denoted by S (or Ω in some books), is a set of points
corresponding to all distinctly different possible outcomes of an
experiment. Each point corresponds to a particular single outcome.
sample point - a single point in a sample space, S

11. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
sample space - denoted by S (or Ω in some books), is a set of points
corresponding to all distinctly different possible outcomes of an
experiment. Each point corresponds to a particular single outcome.
Discrete sample space - one that contains a finite number or
countable infinity of sample points.

12. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
sample space - denoted by S (or Ω in some books), is a set of points
corresponding to all distinctly different possible outcomes of an
experiment. Each point corresponds to a particular single outcome.
Discrete sample space - one that contains a finite number or
countable infinity of sample points.
Continuous sample space - has an infinite number of sample
points.

13. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as a
collection of sample points.

14. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as a
collection of sample points.
Example: Die-tossing Experiment

15. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as a
collection of sample points.
Example: Die-tossing Experiment
A: observe an odd number (A = {1, 3, 5}),

16. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as a
collection of sample points.
Example: Die-tossing Experiment
A: observe an odd number (A = {1, 3, 5}),
B: observe a number less than 5 (B = {1, 2, 3, 4}),

17. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as a
collection of sample points.
Example: Die-tossing Experiment
A: observe an odd number (A = {1, 3, 5}),
B: observe a number less than 5 (B = {1, 2, 3, 4}),
C: observe a 2 or a 3 (C = {2, 3}),

18. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as a
collection of sample points.
Example: Die-tossing Experiment
A: observe an odd number (A = {1, 3, 5}),
B: observe a number less than 5 (B = {1, 2, 3, 4}),
C: observe a 2 or a 3 (C = {2, 3}),
E1 : observe a 1 (E1 = {1}),

19. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as a
collection of sample points.
Example: Die-tossing Experiment
A: observe an odd number (A = {1, 3, 5}),
B: observe a number less than 5 (B = {1, 2, 3, 4}),
C: observe a 2 or a 3 (C = {2, 3}),
E1 : observe a 1 (E1 = {1}),
E6 : observe a 6 (E6 = {6})

20. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as a
collection of sample points.
Example: Die-tossing Experiment
A: observe an odd number (A = {1, 3, 5}),
B: observe a number less than 5 (B = {1, 2, 3, 4}),
C: observe a 2 or a 3 (C = {2, 3}),
E1 : observe a 1 (E1 = {1}),
E6 : observe a 6 (E6 = {6})
Each of these 5 events is a specific collection of sample points.

21. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as a
collection of sample points.
A simple event is one that contains a single sample point. We
may refer to simple events as events that cannot be decomposed.

22. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as a
collection of sample points.
A simple event is one that contains a single sample point. We
may refer to simple events as events that cannot be decomposed.
Probability - a numerical measure of the chance of the occurrence of
an event.

23. Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as a
collection of sample points.
A simple event is one that contains a single sample point. We
may refer to simple events as events that cannot be decomposed.
Probability - a numerical measure of the chance of the occurrence of
an event.
The final step in constructing a probabilistic model for an experiment
with a discrete sample space is to attach a probability to each sample
event.

24. Outline Definitions Approaches Axioms Sample-Point Approach
Approaches to the Assignment of Probability Values

25. Outline Definitions Approaches Axioms Sample-Point Approach
Approaches to the Assignment of Probability Values
Relative Frequency or A Posteriori Approach
The probability value is the relative frequency of the occurrence of
the event over a long-run experiment (over a large number of
repetitions of the experiment).

26. Outline Definitions Approaches Axioms Sample-Point Approach
Approaches to the Assignment of Probability Values
Relative Frequency or A Posteriori Approach
The probability value is the relative frequency of the occurrence of
the event over a long-run experiment (over a large number of
repetitions of the experiment).
number of times the event occurred
P (E) =
number of repetitions of the experiment

27. Outline Definitions Approaches Axioms Sample-Point Approach
Approaches to the Assignment of Probability Values
Relative Frequency or A Posteriori Approach
The probability value is the relative frequency of the occurrence of
the event over a long-run experiment (over a large number of
repetitions of the experiment).
number of times the event occurred
P (E) =
number of repetitions of the experiment
Classical, Theoretical or A Priori Approach
Probability value us based on an experimental model with certain
assumptions

28. Outline Definitions Approaches Axioms Sample-Point Approach
Approaches to the Assignment of Probability Values
Relative Frequency or A Posteriori Approach
The probability value is the relative frequency of the occurrence of
the event over a long-run experiment (over a large number of
repetitions of the experiment).
Classical, Theoretical or A Priori Approach
Probability value us based on an experimental model with certain
assumptions
Subjective Approach
The researcher assigns probability according to his knowledge or
experience on the occurrence of the event. There is no objective way
of prediction of the occurrence of the event under this approach.

29. Outline Definitions Approaches Axioms Sample-Point Approach
Axioms of Probability
For every event E in a sample space S, we assign a numerical value
P(E), known as the probability of E, such that:

30. Outline Definitions Approaches Axioms Sample-Point Approach
Axioms of Probability
For every event E in a sample space S, we assign a numerical value
P(E), known as the probability of E, such that:
1 P(E) 0;

31. Outline Definitions Approaches Axioms Sample-Point Approach
Axioms of Probability
For every event E in a sample space S, we assign a numerical value
P(E), known as the probability of E, such that:
1 P(E) 0;
2 P(S) = 1;

32. Outline Definitions Approaches Axioms Sample-Point Approach
Axioms of Probability
For every event E in a sample space S, we assign a numerical value
P(E), known as the probability of E, such that:
1 P(E) 0;
2 P(S) = 1;
3 If E1 , E2 , . . . form a sequence of pairwise mutually exclusive events
in S (Ei ∩ Ej = ∅, i j), then
∞
P(E1 ∪ E2 ∪ E3 ∪ · · · ) = P(Ai )
i=1

33. Outline Definitions Approaches Axioms Sample-Point Approach
Example
Let A be the event of obtaining a number less than or equal to 3 in
tossing a die.

34. Outline Definitions Approaches Axioms Sample-Point Approach
Example
Let A be the event of obtaining a number less than or equal to 3 in
tossing a die.
Find the probability of A if:

35. Outline Definitions Approaches Axioms Sample-Point Approach
Example
Let A be the event of obtaining a number less than or equal to 3 in
tossing a die.
Find the probability of A if:
1 the die is fair;

36. Outline Definitions Approaches Axioms Sample-Point Approach
Example
Let A be the event of obtaining a number less than or equal to 3 in
tossing a die.
Find the probability of A if:
1 the die is fair;
2 the die is biased such that an odd number is twice as likely to
occur as an even number.

37. Outline Definitions Approaches Axioms Sample-Point Approach
Example
Solution for [1]
First note that S = {1, 2, 3, 4, 5, 6}. Since the die is fair, the probability
for each simple event is equal, say p. That is,
P(1) = P(2) = · · · = P(6) = p.
We further observe that
P(1) + P(2) + · · · + P(6) = 1.
Substituting p to each probability of the simple event, we get
p + p + p + p + p + p = 6p = 1.
1
Thus, p = 6 .

38. Outline Definitions Approaches Axioms Sample-Point Approach
Example
Solution for [1]
The event A = {1, 2, 3}, has therefore a probability:
1 1 1 3
P(A) = P(1) + P(2) + P(3) = + + =
6 6 6 6

39. Outline Definitions Approaches Axioms Sample-Point Approach
Example
Solution for [2]
The sample space of the experiment is still the set S = {1, 2, 3, 4, 5, 6}.
Let p be the probability of each even number to occur and 2p be the
probability of each odd number to occur. That is,
P(2) + P(4) + P(6) =p
P(1) + P(3) + P(5) =2p
Substituting each probability to the simple event, we get
2p + p + 2p + p + 2p + p = 9p = 1.
Thus, p = 1 .
9

40. Outline Definitions Approaches Axioms Sample-Point Approach
Example
Solution for [2]
The event A = {1, 2, 3}, has therefore a probability:
2 1 2 5
P(A) = P(1) + P(2) + P(3) = + + =
9 9 9 9
Not all problems dealing with probability of an event are solvable by
simply using the Axioms of Probability.
Thus, there are 2 ways or approaches known to calculate the
Probability of an Event: the sample-point approach and the
event-composition method.

41. Outline Definitions Approaches Axioms Sample-Point Approach
Steps
Sample-Point Approach on Calculating Probability
Steps:

42. Outline Definitions Approaches Axioms Sample-Point Approach
Steps
Sample-Point Approach on Calculating Probability
Steps:
1 Define the experiment.

43. Outline Definitions Approaches Axioms Sample-Point Approach
Steps
Sample-Point Approach on Calculating Probability
Steps:
1 Define the experiment.
2 List the simple events associated with the experiment and test
each to make certain that they cannot be decomposed. This
defines the sample space, S.

44. Outline Definitions Approaches Axioms Sample-Point Approach
Steps
Sample-Point Approach on Calculating Probability
Steps:
1 Define the experiment.
2 List the simple events associated with the experiment and test
each to make certain that they cannot be decomposed. This
defines the sample space, S.
3 Assign reasonable probabilities to the sample points in S,
making certain that
P(Ei ) = 1
S
.

45. Outline Definitions Approaches Axioms Sample-Point Approach
Steps
Sample-Point Approach on Calculating Probability
Steps:
1 Define the experiment.
2 List the simple events associated with the experiment and test
each to make certain that they cannot be decomposed. This
defines the sample space, S.
3 Assign reasonable probabilities to the sample points in S,
making certain that
P(Ei ) = 1
S
.
4 Define the event of interest, E, as a specific collection of sample
points.

46. Outline Definitions Approaches Axioms Sample-Point Approach
Steps
Sample-Point Approach on Calculating Probability
Steps:
1 Define the experiment.
2 List the simple events associated with the experiment and test
each to make certain that they cannot be decomposed. This
defines the sample space, S.
3 Assign reasonable probabilities to the sample points in S,
making certain that
P(Ei ) = 1
S
.
4 Define the event of interest, E, as a specific collection of sample
points.
5 Find P(E) by summing the probabilities of the sample points in E.

47. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Example 1
Toss a coin 3 times and observe the top face. What is the probability
of observing exactly 2 heads, assuming the coin is fair?

48. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Solution
1 Experiment: Tossing a fair coin 3 times.

49. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Solution
1 Experiment: Tossing a fair coin 3 times.
2 List of simple events:
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

50. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Solution
1 Experiment: Tossing a fair coin 3 times.
2 List of simple events:
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
3 Assignment of probability to each sample points:
1
P(Ei ) = , i = 1, 2, . . . , 8.
8

51. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Solution
1 Experiment: Tossing a fair coin 3 times.
2 List of simple events:
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
3 Assignment of probability to each sample points:
1
P(Ei ) = , i = 1, 2, . . . , 8.
8
4 Define event of interest: Let A be the event that 2 heads will
appear after tossing the coin 3 times.

52. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Solution
1 Experiment: Tossing a fair coin 3 times.
2 List of simple events:
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
3 Assignment of probability to each sample points:
1
P(Ei ) = , i = 1, 2, . . . , 8.
8
4 Define event of interest: Let A be the event that 2 heads will
appear after tossing the coin 3 times.
5 Find P(A):
1 1 1 3
P(A) = P(HHT) + P(HTH) + P(THH) = + + =
8 8 8 8

53. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Example 2
Patients arriving at a hospital outpatient clinic can select any of three
service counters. Physicians are randomly assigned to the stations
and the patients have no station preference. Three patients arrived at
the clinic and their selection is observed. Find the probability that
each station receives a patient.

54. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Solution
1 Experiment: Assigning patients to service counters.

55. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Solution
1 Experiment: Assigning patients to service counters.
2 Let (a, b, c) be the ordered triple where a, b, c ∈ {1, 2, 3}. That is,
each patient could be assigned to any of the service counter 1,2
and 3. Furthermore, |S| = 33 = 27.

56. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Solution
1 Experiment: Assigning patients to service counters.
2 Let (a, b, c) be the ordered triple where a, b, c ∈ {1, 2, 3}. That is,
each patient could be assigned to any of the service counter 1,2
and 3. Furthermore, |S| = 33 = 27.
3 Since each simple events are likely to occur, then
1 1
P(Ei ) = = , ∀i = 1, 2, . . . , 27
|S| 27

57. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Solution
1 Experiment: Assigning patients to service counters.
2 Let (a, b, c) be the ordered triple where a, b, c ∈ {1, 2, 3}. That is,
each patient could be assigned to any of the service counter 1,2
and 3. Furthermore, |S| = 33 = 27.
3 Since each simple events are likely to occur, then
1 1
P(Ei ) = = , ∀i = 1, 2, . . . , 27
|S| 27
4 Define event of interest: Let B be the event that each station
receives a patient.

58. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Solution
1 Experiment: Assigning patients to service counters.
2 Let (a, b, c) be the ordered triple where a, b, c ∈ {1, 2, 3}. That is,
each patient could be assigned to any of the service counter 1,2
and 3. Furthermore, |S| = 33 = 27.
3 Since each simple events are likely to occur, then
1 1
P(Ei ) = = , ∀i = 1, 2, . . . , 27
|S| 27
4 Define event of interest: Let B be the event that each station
receives a patient.
5 P(B) = P((1, 2, 3)) + P((1, 3, 2)) + · · · + P((3, 2, 1))
1 1 1 6
= 27 + 27 + · · · + 27 = 27

59. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Example 3
Four cards are drawn from a standard deck of 52 cards. What is the
probability that the cards drawn are:
1 of the same suit;
2 of the same color;
3 of the same type.

60. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Assignment 1
Write your STEP-BY-STEP solution in a 1/2 sheet of yellow paper.

61. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Assignment 1
Write your STEP-BY-STEP solution in a 1/2 sheet of yellow paper.
A box contains seven laptops. Unknown to the purchaser, three are
defective. Two of the seven are selected for thorough testing and then
classified as defective or nondefective.

62. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Assignment 1
Write your STEP-BY-STEP solution in a 1/2 sheet of yellow paper.
A box contains seven laptops. Unknown to the purchaser, three are
defective. Two of the seven are selected for thorough testing and then
classified as defective or nondefective.
(i) Find the probability of the event A that the selection includes no
defective.

63. Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Assignment 1
Write your STEP-BY-STEP solution in a 1/2 sheet of yellow paper.
A box contains seven laptops. Unknown to the purchaser, three are
defective. Two of the seven are selected for thorough testing and then
classified as defective or nondefective.
(i) Find the probability of the event A that the selection includes no
defective.
(ii) Find the probability of the event B that the selection includes
exactly one defective.