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CHAPTER 6
Section 6.2 Part 2 – Probability Rules
PROBABILITY RULES
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Rule 1 :
The probability 𝑃 𝐴 of any event A satisfies:
0≤𝑃 𝐴 ≤1
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Rule 2 :
If S is the sample space in probability model, then:
𝑃 𝑆 =1
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Rule 3 :
The compliment of any event A is the event that A does not
occur, written as 𝐴𝑐 . The compliment rule states that:
𝑃 𝐴𝑐 = 1 − 𝑃(𝐴)
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The compliment of an event can also be represented as:
𝑃(𝐴′ ) or 𝑃(𝐴)
PROBABILITY RULES
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Rule 4 :
Two events A and B are disjoint (also called
mutually exclusive) if they have no outcomes in
common and so can never occur simultaneously.
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If A and B are disjoint then,
𝑃 𝐴 or 𝐵 = 𝑃 𝐴 + 𝑃(𝐵)
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This is the addition rule 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.
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If A and B are independent, then
𝑃 𝐴 and 𝐵 = 𝑃 𝐴 𝑃(𝐵)
 This is the multiplication rule 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 FOR DISJOINT EVENTS
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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
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The compliment 𝐴𝑐 in the diagram below
contains exactly the outcomes not in A.
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Note that we could write 𝐴 ∪ 𝐴𝑐 = 𝑆 and 𝐴 ∩ 𝐴𝑐 =∅.
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See examples 6.8 and 6.9 on p.344-345
VENN DIAGRAM OF INDEPENDENT EVENTS
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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
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Be careful not to confuse disjointness with independence
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Recall that disjoint events (or mutually exclusive events) tell
us that if event A occurs that event B cannot occur
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With independent events the outcome of one trial must not
influence the outcome of any other
For example:
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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.
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Homework: p.348-358 #’s 19, 20, 23, 26 (ignore
Benfords Law, just use the probabilities), 28, 30,
31, & 41