4.3: Some Elementary Probability Rules Flashcards
What is the complement of an event in probability theory?
The complement of an event A, denoted as Ā, is the event consisting of all sample space outcomes that do not correspond to the occurrence of A.
What does the probability P(Ā) represent, and how is it related to the event A?
The probability P(Ā) represents the probability that the complement event Ā will occur, which means that event A will not occur.
It is related to event A in that P(Ā) is the probability of all outcomes not in A, essentially the opposite of event A.
Explain the relationship between event A, its complement Ā, and the total probability of the sample space.
In any probability situation, either event A or its complement Ā must occur. Therefore, the sum of the probabilities of these two events is equal to
P(A)+P(A)=1
This relationship indicates that the probability of event A occurring plus the probability of event A not occurring (its complement) equals the total probability of the sample space, which is always 1.
The Compement of an Event (Diagram)
What is The Rule Of Complements?
What is the intersection of two events, and how is it denoted in probability theory?
The intersection of two events A and B, denoted as A ∩ B, is the event that occurs if both events A and B simultaneously occur.
P(A ∩ B) represents the probability that both A and B will occur simultaneously.
What is the union of two events, and how is it denoted in probability theory?
The union of two events A and B, denoted as A ∪ B, is the event that occurs if either A or B (or both) occur.
P(A ∪ B) represents the probability that either event A or event B (or both) will occur.
Diagram depecting events
Picture: Two mutually exclusive events
What is the Addition Rule in probability, and how is it applied to events A and B?
The Addition Rule in probability states that the probability that event A or event B (or both) will occur is calculated as the sum of the probabilities of A and B minus the probability of their intersection:
P(A∪B)=P(A)+P(B)−P(A∩B)
It is used to find the probability of either event A or event B (or both) happening.
What are mutually exclusive events, and what is the probability relationship between them?
Mutually exclusive events A and B are events that have no sample space outcomes in common.
In this case, events A and B cannot occur simultaneously, and the probability of both events occurring is zero:
P(A∩B)=0
When calculating the probability of the union of two events (A or B), why is it necessary to subtract the probability of their intersection (A ∩ B)?
When calculating the probability of the union of two events (A or B), it is necessary to subtract the probability of their intersection (A ∩ B) because the intersection represents the outcomes that are counted twice when simply adding the probabilities of A and B.
Subtracting the intersection corrects for the double-counting of outcomes that are common to both events.
What is the Addition Rule for Two Mutually Exclusive Events, and when is it applicable?
The Addition Rule for Two Mutually Exclusive Events states that if events A and B are mutually exclusive (they have no sample space outcomes in common), the probability of either event A or event B occurring is simply the sum of their individual probabilities:
P(A∪B)=P(A)+P(B)
This rule applies when the events are mutually exclusive, meaning that they cannot occur simultaneously.
How is the Addition Rule extended to an arbitrary group of events, and what is the formula for it?
For an arbitrary group of events A 1, A 2,…,A N
the probability that at least one of the events occurs (the union of these events) is denoted as
P(A1∪ A2 ∪…∪ AN).
Although there is a formula for this probability, it can be quite complicated and is not presented here.
Instead, sample spaces can sometimes be used to reason out the probability.
If the events are mutually exclusive, there is a simplified formula for the probability that at least one of the events will occur, known as the Addition Rule for N Mutually Exclusive Events:
P(A1 ∪ A2 ∪…∪ AN)=P(A1)+P(A2)+…+P(AN)
This formula applies when the events
A 1, A2,…,AN
are mutually exclusive, meaning no two of the events have any sample space outcomes in common.
