Transcript P. STATISTICS LESSON 6 – 2 (DAY2)

A. P. STATISTICS
LESSON 6 – 2 (DAY2)
PROBABILITY RULES
ESSENTIAL QUESTION: What
are the probability rules and how
are they used to solve problems?
Objectives :
 To become familiar with the probability
rules.
 To use the probability rules to solve
problems in which they may be used.

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 does not occur, written as Ac . The compliment
rule states that
P(Ac) = 1 – 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,
P(A or B) = P(A) + P(B)
This is the addition rule for disjoint events.
Set Notation
A U B – read “ A union B” is the set of all
outcomes that are either in A or B’
Empty event – The event that has no outcomes in
it. (ø)
If two events A and B are disjoint (mutually
exclusively), we can write A ∩ B = ø, read
“intersect B is empty.”
A picture like that shows the Sample space S as a
rectangular area and events as areas within S is
called a Venn diagram.
Venn Diagram
The events A and B are disjoint because
they do not overlap;
S
A
B
Compliment Ac
The compliment Ac contains exactly the
outcomes not in A. Note that we could
write A U Ac = A ∩ B = ø.
A
Ac
Example 6.8
 Page 344
Example 6.9
 Page 344 probabilities for rolling dice.
Assigning probabilities: finite
number of outcomes
 PROBABILITIES IN A FINITE SAMPLE
SPACE
Assign a probability to each individual
outcome. These probabilities must be
between numbers 0 and 1 must have sum
1.
The probability of any event is the sum of
probabilities of the outcomes making up
the event.
Benford’s Law
 Page 345 example 6.10
Used in accounting as a test for faked
numbers in tax returns, payment records,
invoices, expense account claims, and
many other settings often display patterns
that aren’t present in legitimate records.
Usually applies to first digits.
Assigning probabilities: equally
likely outcomes
 Assigning correct probabilities to individual
outcomes often requires long observation
of the random phenomenon. In some
special circumstances, however, we are
willing to assume that individual outcomes
are equally likely because of some
balance in the phenomenon.
Equally likely Outcomes
If a random phenomenon has k possible
outcomes, all equally likely, then each
individual outcome has probability 1/k.
The probability of any event A is
P(A) = count of outcomes in A
Count of outcomes in S
= count of outcomes in A
k
The Multiplication Rule for
Independent 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,
P(A and B) = P(A)P(B)
This is the multiplication rule for independent
events.
Venn diagram of independent
events
A and B
A
B
Independent and Disjoint
Disjoint – Mutually exclusive
Independent – The outcome of one trial must not
influence the outcome of any other.
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.