Transcript Sec. 6.3 Part 1 PowerPoint

CHAPTER 6
Section 6.3 Part 1 – Probability Rules
PROBABILITY RULES
Recall the following probability rules from sec. 6.2:











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 does not occur, written as 𝐴𝑐 . The
compliment rule states that:
𝑃 𝐴𝑐 = 1 − 𝑃(𝐴)
 The compliment of an event can also be represented as:
𝑃(𝐴′ ) or 𝑃(𝐴)
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.

If A and B are disjoint then,
𝑃 𝐴 or 𝐵 = 𝑃 𝐴 + 𝑃(𝐵)
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)
ADDITION RULE FOR DISJOINT EVENTS


The general probability rules can be extended to more
than two events.
Addition rule for disjoint events:

If events A, B, and C are disjoint in the sense that no two
have any outcomes in common, then
𝑃 one or more of 𝐴, 𝐵, 𝐶 = 𝑃 𝐴 + 𝑃 𝐵 + 𝑃(𝐶)

This rule extends to any number of disjoint events.
GENERAL ADDITION RULE FOR UNIONS OF
TWO EVENTS



The union (represented by the word “or”) of any collection
of events is the event that at least one of the collection
occurs.
If events are not disjoint, they can occur simultaneously.
The following diagram suggests that the outcomes common
to both are counted twice when adding the probabilities, so
we must subtract the probability once.
GENERAL ADDITION RULE FOR UNIONS OF
TWO EVENTS

The following is the addition rule for the union of any two
events, disjoint or not:

For any two events A and B,
𝑃 𝐴 or 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 and 𝐵)
or
𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)

Note – even though the event {A and B} cannot happen for
disjoint events, it is still included in the formula since it’s an
empty event ∅ that wouldn’t affect the probabilities anyway.
EXAMPLE

To improve tourism between France and the U.S., the two governments form a
committee consisting of 20 people: 2 American men, 4 French men, 6 American
Women, and 8 French women. If you meet one of these people at random, what is
the probability that the person will be either a woman or a French person?
𝑃 Woman or French =
probability
of a woman

14
20
+
12
20
−
probability of
a French person
Also see example 6.17 on p.362-363
8
20
=
18
20
or
probability of
a French woman
9
10
JOINT PROBABILITY

A joint event is a simultaneous occurrence of two events.

The probability of a joint event is called a joint probability.

Another way to work with joint events (other than a Venn diagram) is to
use tables.
CONDITIONAL PROBABILITY

Conditional probability – gives the probability of
one event under the condition that we know
another event.

Represented by the notation 𝑃(𝐴 𝐵) and read as “the
probability of A given the information B”

Ex. – the probability that the next card dealt is an ace given
that exactly one of the four visible cards is an ace.

3
1
𝑃(ace 1 ace in 4 visible cards) = 48 = 16
GENERAL MULTIPLICATION RULE FOR
ANY TWO EVENTS

The probability that both of two events A and B
happen together can be found by:
𝑃 𝐴 and 𝐵 = 𝑃(𝐴) × 𝑃(𝐵 𝐴)



Here 𝑃(𝐵 𝐴) is the conditional probability that B
occurs given the information that A occurs.
In words, this rule says that for both of two
events to occur, first one must occur and then,
given that the first event has occurred, the
second must occur.
See example 6.20 on p.368
DEFINITION OF CONDITIONAL PROBABILITY

If you know P(A) and P(A and B) you can
rearrange the multiplication rule to produce a
definition of the conditional probability in terms
of unconditional probabilities:


When P(A) > 0, the conditional probability of B given
A is:
𝑃(𝐴 and 𝐵)
𝑃 𝐵𝐴 =
𝑃(𝐴)
See example 6.21 on p.369

Homework P.364-371 #’s 46-51, 53, 56, 58, 59, &
61