Download Section 6.2 Part 2 – Probability Rules

Section 6.2 Part 2 – Probability Rules
General Probability Rules:
Rule 1 :

The probability 𝑃(𝐴) of any event A satisfies: 0 ≤ 𝑃(𝐴) ≤ 1
Rule 2 :

If S is the sample space in probability model, then: 𝑃(𝑆) = 1
Rule 3 :

The compliment of any event A is the event that A _________________, written as _______. The compliment
rule states that: 𝑃(𝐴𝑐 ) = 1 − 𝑃(𝐴)

The compliment of an event can also be represented as: 𝑃(𝐴′ ) or 𝑃(𝐴̅)
Rule 4 :
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Two events A and B are disjoint (also called mutually exclusive) if they have no outcomes in common and so can
never occur simultaneously.

If A and B are disjoint then, 𝑃(𝐴 or 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)

This is the ____________________ for disjoint events.
Rule 5 :

Two events A and B are independent if knowing that one occurs does not change the probability that the other
occurs.

If A and B are independent, then 𝑃(𝐴 and 𝐵) = 𝑃(𝐴)𝑃(𝐵)

This is the ____________________ for independent events.
Set Notation
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𝐴 ∪ 𝐵 – read “A union B” is the set of all outcomes that are either in A or B.

𝐴 ∩ 𝐵 – read “A intersect B” is the set of all outcomes that are in A and B.

Empty event ∅– The event that has no outcomes in it.

If two events A and B are disjoint (mutually exclusively), we can write 𝐴 ∩ 𝐵 = ∅, read “A intersect B is empty.”
Venn Diagram of Disjoint Events

The following picture shows the sample space S as a rectangular area and events as areas within S is called a
Venn diagram.

The events A and B are disjoint because they do not overlap; that is, they have no outcomes in common.
Compliment 𝑨𝒄

The compliment 𝐴𝑐 in the diagram below contains exactly the outcomes not in A.

Note that we could write 𝐴 ∪ 𝐴𝑐 = 𝑆 and 𝐴 ∩ 𝐴𝑐 = ∅.

See examples 6.8 and 6.9 on p.344-345
Venn Diagram of Independent Events

Suppose that you toss a coin twice. You are counting heads so two events of interest are:
𝐴 = first toss is a head
𝐵 = second toss is a head

The events A and B are not disjoint. They occur together whenever both tosses give heads.

The Venn diagram illustrates the event {A and B} as the overlapping area that is common to both A and B.

See example 6.12 on p.351 and example 6.14 on p.353
Independent and Disjoint


Be careful not to confuse disjointness with independence

Recall that disjoint events (or mutually exclusive events) tell us that if event A occurs that event B
cannot occur

With independent events the outcome of one trial must not influence the outcome of any other
For example:


A subject in a study cannot be both male and female, nor can they be aged 20 and 30. A subject could
however be both male and 20, or both female and 30.
Unlike disjointness or compliments, independence cannot be pictured by a Venn diagram, because it involves
the probabilities of the events rather than just the outcomes that make up the events.