STATS
CHAPTER 4
BASIC PROBABILITY
Probability principles:
- Probability distributions
- Mathematical expectation
- Binomial and Poisson distributions
Sample space:
- All possible outcomes of a variable
Events:
Simple event:
- Described by single characteristic
Joint event:
- Described by 2 or more characteristics
Complement of an event A (denoted A’):
- All events that are not part of event A
Probability:
- The numerical value representing the chance, likelihood, or possibility that a certain event
will occur
- Between 0 and 1
Impossible event:
- An event that has no chance of occurring
- Probability = 0
Certain event:
- An event that is sure to occur
- Probability = 1
Mutually Exclusive Events:
- Events that cannot occur simultaneously
- A or B – never both
Collectively Exhaustive Event:
- One of the events must occur
- The set of events covers the entire sample space
- The entire sample space is “exhausted”
Visualizing Probability:
- Venn Diagram:
o A∪B
A or B
o A∩B
A and B
, Approaches to Assessing the Probability of Events:
A Priori:
- Based on prior knowledge
X number of ways ∈which theevent occurs
- Probability of occurrence = =
T total number of possible outcomes
- Assuming all outcomes are equally likely
Empirical Probability:
- Based on observed data
X number of ways ∈which theevent occurs
- Probability of occurrence = =
T total number of possible outcomes
- Assuming all outcomes are equally likely
Subjective Probability:
- Based on combination of an individual’s past experience, personal opinion and analysis of a
situation.
- Differs from person-to-person
Simple Probability:
- Probability of a simple event
number of outcomes satisfying A
- P (A) =
total number of outcomes
Joint Probability:
- Probability of an occurrence of 2 or more events (joint event)
number of outcomes satisfying A∧B
- P (A and B) =
totalnumber of outcomes
Marginal Probability:
- P (A) = P (A and B1) + P (A and B2) + … + P (A and Bk)
- Where B1, B2, …, Bk are k mutually exclusive and collectively exhaustive events.
Marginal & Joint Probabilities in Contingency Table:
Event Event Total
B1 B2
A1 P (A1 and B1) P (A1 and B2) P (A1)
A2 P (A2 and B1) P (A2 and B2) P (A2)
Total P (B1) P (B2) 1
Joint Probabilities Marginal (Simple) Probabilities
Probability Summary:
- P is the numerical measure of the likelihood that an event will occur.
- P of any event must be between 0 and 1, inclusively.
0 P (A) 1 for any event A
- The sum of the probabilities of all mutually exclusive and collectively exhaustive events is 1
P(A) + P(B) + P(C) = 1
If A, B, and C are mutually exclusive and
collectively exhaustive
CHAPTER 4
BASIC PROBABILITY
Probability principles:
- Probability distributions
- Mathematical expectation
- Binomial and Poisson distributions
Sample space:
- All possible outcomes of a variable
Events:
Simple event:
- Described by single characteristic
Joint event:
- Described by 2 or more characteristics
Complement of an event A (denoted A’):
- All events that are not part of event A
Probability:
- The numerical value representing the chance, likelihood, or possibility that a certain event
will occur
- Between 0 and 1
Impossible event:
- An event that has no chance of occurring
- Probability = 0
Certain event:
- An event that is sure to occur
- Probability = 1
Mutually Exclusive Events:
- Events that cannot occur simultaneously
- A or B – never both
Collectively Exhaustive Event:
- One of the events must occur
- The set of events covers the entire sample space
- The entire sample space is “exhausted”
Visualizing Probability:
- Venn Diagram:
o A∪B
A or B
o A∩B
A and B
, Approaches to Assessing the Probability of Events:
A Priori:
- Based on prior knowledge
X number of ways ∈which theevent occurs
- Probability of occurrence = =
T total number of possible outcomes
- Assuming all outcomes are equally likely
Empirical Probability:
- Based on observed data
X number of ways ∈which theevent occurs
- Probability of occurrence = =
T total number of possible outcomes
- Assuming all outcomes are equally likely
Subjective Probability:
- Based on combination of an individual’s past experience, personal opinion and analysis of a
situation.
- Differs from person-to-person
Simple Probability:
- Probability of a simple event
number of outcomes satisfying A
- P (A) =
total number of outcomes
Joint Probability:
- Probability of an occurrence of 2 or more events (joint event)
number of outcomes satisfying A∧B
- P (A and B) =
totalnumber of outcomes
Marginal Probability:
- P (A) = P (A and B1) + P (A and B2) + … + P (A and Bk)
- Where B1, B2, …, Bk are k mutually exclusive and collectively exhaustive events.
Marginal & Joint Probabilities in Contingency Table:
Event Event Total
B1 B2
A1 P (A1 and B1) P (A1 and B2) P (A1)
A2 P (A2 and B1) P (A2 and B2) P (A2)
Total P (B1) P (B2) 1
Joint Probabilities Marginal (Simple) Probabilities
Probability Summary:
- P is the numerical measure of the likelihood that an event will occur.
- P of any event must be between 0 and 1, inclusively.
0 P (A) 1 for any event A
- The sum of the probabilities of all mutually exclusive and collectively exhaustive events is 1
P(A) + P(B) + P(C) = 1
If A, B, and C are mutually exclusive and
collectively exhaustive
