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```Chapter 15
Probability Rules!
.

When two events A and B are disjoint, we can
use the addition rule for disjoint events –Chap 14:

However, when our events are not disjoint, this
earlier addition rule will double count the
probability of both A and B occurring. Thus, we
need the
Slide 15- 2

The following Venn diagram shows a situation in
which we would use the general addition rule:


For any two events A and B,
Slide 15- 3
It Depends…


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Back in Chap 3, we looked at contingency tables
When we want the probability of an event from a
conditional distribution, we write
and
pronounce it “the probability of B given A.”
A probability that takes into account a given
condition is called a
Slide 15- 4
It Depends… (cont.)

To find the probability of the event B given the
event A, we restrict our attention to the outcomes
in A. We then find in what fraction of those
outcomes B also occurred

Note: P(A) cannot equal 0, since we know that A
has occurred
Slide 15- 5
The General Multiplication Rule

When two events A and B are independent, we
can use the multiplication rule for independent
events from Chap 14:

However, when our events are not independent,
this earlier multiplication rule does not work.
Thus, we need the
Slide 15- 6
The General Multiplication Rule (cont.)


We encountered the general multiplication rule in
the form of conditional probability
Rearranging the equation in the definition for
conditional probability, we get the
- For any two events A and B,
or
Slide 15- 7
Independence


Independence of two events means that the
outcome of one event does not influence the
probability of the other
With our new notation for conditional
probabilities, we can now formalize this definition:
- Events A and B are
whenever
. (Equivalently, events A and
B are independent whenever P(A|B) = P(A).)
Slide 15- 8
Independent ≠ Disjoint

Disjoint events cannot be independent! Why not?
- Since we know that disjoint events have no outcomes in
common, knowing that one occurred means the other
didn’t
- Thus, the probability of the second occurring changed
based on our knowledge that the first occurred
- It follows, then, that the two events are not independent

A common error is to treat disjoint events as if
they were independent
Slide 15- 9
Depending on Independence


It’s much easier to think about independent
events than to deal with conditional probabilities
 It seems that most people’s natural intuition for
probabilities breaks down when it comes to
conditional probabilities
Don’t fall into this trap: whenever you see
probabilities multiplied together, stop and ask
whether you think they are really independent
Slide 15- 10
What Can Go Wrong?
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Don’t use a simple probability rule where a
general rule is appropriate:

Don’t assume that two events are
independent or disjoint without checking that
they are
Don’t confuse “disjoint” with “independent.”
Slide 15- 11
What have we learned?

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The probability rules from Chapter 14 only work
in special cases—when events are disjoint or
independent
We now know the General Addition Rule and
General Multiplication Rule
We also know about conditional probabilities
Venn diagrams and tables help organize our
We now know more about independence
Slide 15- 12
Example
Boy
Girl
Total
117
130
247
50
91
141
60
30
90
227
251
478
Slide 15- 13
Example
If we choose a student at random, what’s the
probability of choosing a girl?
# girls
P( girl ) 
total #
251
P( girl ) 
 0.525
478
Slide 15- 14
Example
If we choose a student at random, what’s the
probability of choosing a student who is a girl and
has a goal of being a popular?
# girls with popular goal
P(girl and popular ) 
total #
91
P(girl and popular ) 
 0.190
478
Slide 15- 15
Example
If we choose a student at random, what’s the
probability of choosing a student who has sports
as a goal?
# sports
P( sports ) 
total #
90
P ( sports ) 
 0.188
478
Slide 15- 16
Example

If we choose a student at random, what’s the
probability of choosing a student who’s goal is
getting good grades or being good at sports?
 Grades and sports are disjoint!!
247 90

478 478
P(grades or sports) 0.517  0.188  0.705
Slide 15- 17
Example

If we choose a student at random, what’s the
probability of choosing a student who’s goal is
being good at sports or a female?
 Grades and female are NOT disjoint!!
P(sportsor female) P (sports) P (female)- P (sportsand female)
90 251 30
P (sportsor female)


478 478 478
P(sportsor female) 0.188 0.525- 0.063 0.65
Slide 15- 18
Example
If we choose a student at random, what’s the
probability of choosing a girl who is interested in
sports?
# sportsand girls
P( sports | girl) 
# girls
30
P( sports | girl ) 
 0.120
251
Slide 15- 19
Example

Is the probability of having good grades as a goal
independent of the sex of the responding
student?