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Outline Definitions Approaches Axioms Sample-Point Approach
1. Sets and Probability1.3 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
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 ProbabilityStepsExamples
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 ProbabilityStepsExamples
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 ProbabilityStepsExamples
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 ProbabilityStepsExamples
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
experiment - the process of making an observation.
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 distinctlydifferent observable outcomes.
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 distinctlydifferent observable outcomes.
We are interested in experiments that generate outcomes which vary inrandom manner and cannot be predicted with certainty.
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 pointscorresponding to all distinctly different possible outcomes of anexperiment. Each point corresponds to a particular single outcome.
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 pointscorresponding to all distinctly different possible outcomes of anexperiment. Each point corresponds to a particular single outcome.
sample point - a single point in a sample space, S
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
sample space - denoted by S (or Ω in some books), is a set of pointscorresponding to all distinctly different possible outcomes of anexperiment. Each point corresponds to a particular single outcome.
Discrete sample space - one that contains a finite number orcountable infinity of sample points.
Continuous sample space - has an infinite number of samplepoints.
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
sample space - denoted by S (or Ω in some books), is a set of pointscorresponding to all distinctly different possible outcomes of anexperiment. Each point corresponds to a particular single outcome.
Discrete sample space - one that contains a finite number orcountable infinity of sample points.Continuous sample space - has an infinite number of samplepoints.
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as acollection of sample points.
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as acollection 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)
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as acollection 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)
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as acollection 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)
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as acollection 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)
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as acollection 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)
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as acollection 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)
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as acollection 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.
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as acollection of sample points.
A simple event is one that contains a single sample point. Wemay refer to simple events as events that cannot be decomposed.
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as acollection of sample points.
A simple event is one that contains a single sample point. Wemay refer to simple events as events that cannot be decomposed.
Probability - a numerical measure of the chance of the occurrence ofan event.
Outline Definitions Approaches Axioms Sample-Point Approach
Definitions
event - any subset of the sample space, S. It can also be viewed as acollection of sample points.
A simple event is one that contains a single sample point. Wemay refer to simple events as events that cannot be decomposed.
Probability - a numerical measure of the chance of the occurrence ofan event.
The final step in constructing a probabilistic model for an experimentwith a discrete sample space is to attach a probability to each sampleevent.
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 ofthe event over a long-run experiment (over a large number ofrepetitions of the experiment).
Classical, Theoretical or A Priori Approach
Probability value us based on an experimental model with certainassumptions
Subjective Approach
The researcher assigns probability according to his knowledge orexperience on the occurrence of the event. There is no objective wayof prediction of the occurrence of the event under this approach.
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 ofthe event over a long-run experiment (over a large number ofrepetitions of the experiment).
Classical, Theoretical or A Priori Approach
Probability value us based on an experimental model with certainassumptions
Subjective Approach
The researcher assigns probability according to his knowledge orexperience on the occurrence of the event. There is no objective wayof prediction of the occurrence of the event under this approach.
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 ofthe event over a long-run experiment (over a large number ofrepetitions of the experiment).
P (E) =number of times the event occurred
number of repetitions of the experiment
Classical, Theoretical or A Priori Approach
Probability value us based on an experimental model with certainassumptions
Subjective Approach
The researcher assigns probability according to his knowledge orexperience on the occurrence of the event. There is no objective wayof prediction of the occurrence of the event under this approach.
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 ofthe event over a long-run experiment (over a large number ofrepetitions of the experiment).
P (E) =number of times the event occurred
number of repetitions of the experiment
Classical, Theoretical or A Priori Approach
Probability value us based on an experimental model with certainassumptions
Subjective Approach
The researcher assigns probability according to his knowledge orexperience on the occurrence of the event. There is no objective wayof prediction of the occurrence of the event under this approach.
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 ofthe event over a long-run experiment (over a large number ofrepetitions of the experiment).
Classical, Theoretical or A Priori Approach
Probability value us based on an experimental model with certainassumptions
Subjective Approach
The researcher assigns probability according to his knowledge orexperience on the occurrence of the event. There is no objective wayof prediction of the occurrence of the event under this approach.
Outline Definitions Approaches Axioms Sample-Point Approach
Axioms of Probability
For every event E in a sample space S, we assign a numerical valueP(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 ∪ · · · ) =∞∑
i=1
P(Ai)
Outline Definitions Approaches Axioms Sample-Point Approach
Axioms of Probability
For every event E in a sample space S, we assign a numerical valueP(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 ∪ · · · ) =∞∑
i=1
P(Ai)
Outline Definitions Approaches Axioms Sample-Point Approach
Axioms of Probability
For every event E in a sample space S, we assign a numerical valueP(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 eventsin S (Ei ∩ Ej = ∅, i , j), then
P(E1 ∪ E2 ∪ E3 ∪ · · · ) =∞∑
i=1
P(Ai)
Outline Definitions Approaches Axioms Sample-Point Approach
Axioms of Probability
For every event E in a sample space S, we assign a numerical valueP(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 ∪ · · · ) =∞∑
i=1
P(Ai)
Outline Definitions Approaches Axioms Sample-Point Approach
Example
Let A be the event of obtaining a number less than or equal to 3 intossing 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.
Outline Definitions Approaches Axioms Sample-Point Approach
Example
Let A be the event of obtaining a number less than or equal to 3 intossing 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.
Outline Definitions Approaches Axioms Sample-Point Approach
Example
Let A be the event of obtaining a number less than or equal to 3 intossing 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 tooccur as an even number.
Outline Definitions Approaches Axioms Sample-Point Approach
Example
Let A be the event of obtaining a number less than or equal to 3 intossing 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.
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 probabilityfor 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.
Thus, p = 16 .
Outline Definitions Approaches Axioms Sample-Point Approach
Example
Solution for [1]
The event A = 1, 2, 3, has therefore a probability:
P(A) = P(1) + P(2) + P(3) =16+
16+
16=
36
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 theprobability of each odd number to occur. That is,
P(2) + P(4) + P(6) =pP(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 = 19 .
Outline Definitions Approaches Axioms Sample-Point Approach
Example
Solution for [2]
The event A = 1, 2, 3, has therefore a probability:
P(A) = P(1) + P(2) + P(3) =29+
19+
29=
59
Not all problems dealing with probability of an event are solvable bysimply using the Axioms of Probability.
Thus, there are 2 ways or approaches known to calculate theProbability of an Event: the sample-point approach and theevent-composition method.
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. Thisdefines the sample space, S.
3 Assign reasonable probabilities to the sample points in S,making certain that ∑
S
P(Ei) = 1
.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.
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 testeach to make certain that they cannot be decomposed. Thisdefines the sample space, S.
3 Assign reasonable probabilities to the sample points in S,making certain that ∑
S
P(Ei) = 1
.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.
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. Thisdefines the sample space, S.
3 Assign reasonable probabilities to the sample points in S,making certain that ∑
S
P(Ei) = 1
.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.
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. Thisdefines the sample space, S.
3 Assign reasonable probabilities to the sample points in S,making certain that ∑
S
P(Ei) = 1
.
4 Define the event of interest, E, as a specific collection of samplepoints.
5 Find P(E) by summing the probabilities of the sample points in E.
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. Thisdefines the sample space, S.
3 Assign reasonable probabilities to the sample points in S,making certain that ∑
S
P(Ei) = 1
.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.
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. Thisdefines the sample space, S.
3 Assign reasonable probabilities to the sample points in S,making certain that ∑
S
P(Ei) = 1
.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.
Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Example 1
Toss a coin 3 times and observe the top face. What is the probabilityof observing exactly 2 heads, assuming the coin is fair?
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:
P(Ei) =18
, i = 1, 2, . . . , 8.
4 Define event of interest: Let A be the event that 2 heads willappear after tossing the coin 3 times.
5 Find P(A):
P(A) = P(HHT) + P(HTH) + P(THH) =18+
18+
18=
38
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:
P(Ei) =18
, i = 1, 2, . . . , 8.
4 Define event of interest: Let A be the event that 2 heads willappear after tossing the coin 3 times.
5 Find P(A):
P(A) = P(HHT) + P(HTH) + P(THH) =18+
18+
18=
38
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:
P(Ei) =18
, i = 1, 2, . . . , 8.
4 Define event of interest: Let A be the event that 2 heads willappear after tossing the coin 3 times.
5 Find P(A):
P(A) = P(HHT) + P(HTH) + P(THH) =18+
18+
18=
38
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:
P(Ei) =18
, i = 1, 2, . . . , 8.
4 Define event of interest: Let A be the event that 2 heads willappear after tossing the coin 3 times.
5 Find P(A):
P(A) = P(HHT) + P(HTH) + P(THH) =18+
18+
18=
38
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:
P(Ei) =18
, i = 1, 2, . . . , 8.
4 Define event of interest: Let A be the event that 2 heads willappear after tossing the coin 3 times.
5 Find P(A):
P(A) = P(HHT) + P(HTH) + P(THH) =18+
18+
18=
38
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:
P(Ei) =18
, i = 1, 2, . . . , 8.
4 Define event of interest: Let A be the event that 2 heads willappear after tossing the coin 3 times.
5 Find P(A):
P(A) = P(HHT) + P(HTH) + P(THH) =18+
18+
18=
38
Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Example 2
Patients arriving at a hospital outpatient clinic can select any of threeservice counters. Physicians are randomly assigned to the stationsand the patients have no station preference. Three patients arrived atthe clinic and their selection is observed. Find the probability thateach station receives a patient.
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 1,2and 3. Furthermore, |S| = 33 = 27.
3 Since each simple events are likely to occur, then
P(Ei) =1|S|
=1
27, ∀i = 1, 2, . . . , 27
4 Define event of interest: Let B be the event that each stationreceives a patient.
5 P(B) = P((1, 2, 3)) + P((1, 3, 2)) + · · ·+ P((3, 2, 1))= 1
27 + 127 + · · ·+ 1
27 = 627
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 1,2and 3. Furthermore, |S| = 33 = 27.
3 Since each simple events are likely to occur, then
P(Ei) =1|S|
=1
27, ∀i = 1, 2, . . . , 27
4 Define event of interest: Let B be the event that each stationreceives a patient.
5 P(B) = P((1, 2, 3)) + P((1, 3, 2)) + · · ·+ P((3, 2, 1))= 1
27 + 127 + · · ·+ 1
27 = 627
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 1,2and 3. Furthermore, |S| = 33 = 27.
3 Since each simple events are likely to occur, then
P(Ei) =1|S|
=1
27, ∀i = 1, 2, . . . , 27
4 Define event of interest: Let B be the event that each stationreceives a patient.
5 P(B) = P((1, 2, 3)) + P((1, 3, 2)) + · · ·+ P((3, 2, 1))= 1
27 + 127 + · · ·+ 1
27 = 627
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 1,2and 3. Furthermore, |S| = 33 = 27.
3 Since each simple events are likely to occur, then
P(Ei) =1|S|
=1
27, ∀i = 1, 2, . . . , 27
4 Define event of interest: Let B be the event that each stationreceives a patient.
5 P(B) = P((1, 2, 3)) + P((1, 3, 2)) + · · ·+ P((3, 2, 1))= 1
27 + 127 + · · ·+ 1
27 = 627
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 1,2and 3. Furthermore, |S| = 33 = 27.
3 Since each simple events are likely to occur, then
P(Ei) =1|S|
=1
27, ∀i = 1, 2, . . . , 27
4 Define event of interest: Let B be the event that each stationreceives a patient.
5 P(B) = P((1, 2, 3)) + P((1, 3, 2)) + · · ·+ P((3, 2, 1))= 1
27 + 127 + · · ·+ 1
27 = 627
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 1,2and 3. Furthermore, |S| = 33 = 27.
3 Since each simple events are likely to occur, then
P(Ei) =1|S|
=1
27, ∀i = 1, 2, . . . , 27
4 Define event of interest: Let B be the event that each stationreceives a patient.
5 P(B) = P((1, 2, 3)) + P((1, 3, 2)) + · · ·+ P((3, 2, 1))= 1
27 + 127 + · · ·+ 1
27 = 627
Outline Definitions Approaches Axioms Sample-Point Approach
Examples
Example 3
Four cards are drawn from a standard deck of 52 cards. What is theprobability that the cards drawn are:
1 of the same suit;2 of the same color;3 of the same type.
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 aredefective. Two of the seven are selected for thorough testing and thenclassified as defective or nondefective.
(i) Find the probability of the event A that the selection includes nodefective.
(ii) Find the probability of the event B that the selection includesexactly one defective.
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 aredefective. Two of the seven are selected for thorough testing and thenclassified as defective or nondefective.
(i) Find the probability of the event A that the selection includes nodefective.
(ii) Find the probability of the event B that the selection includesexactly one defective.
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 aredefective. Two of the seven are selected for thorough testing and thenclassified as defective or nondefective.
(i) Find the probability of the event A that the selection includes nodefective.
(ii) Find the probability of the event B that the selection includesexactly one defective.
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 aredefective. Two of the seven are selected for thorough testing and thenclassified as defective or nondefective.
(i) Find the probability of the event A that the selection includes nodefective.
(ii) Find the probability of the event B that the selection includesexactly one defective.