1. Chapter 15 Probability Rules!
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Recall That… For any random phenomenon, each trial generates an .
An is any set or collection of outcomes.
The collection of all possible outcomes is called the , denoted
outcome
event
sample space S

3. Events
When outcomes are , probabilities for events are easy to find just by counting.
When the k possible outcomes are equally likely, each has a probability of
For any event A that is made up of equally likely outcomes,
equally likely
1/k

4. The First Three Rules for Working with Probability Rules
Make a picture.

5. Picturing Probabilities
The most common kind of picture to make is called a diagram.
We will see diagrams in practice shortly…
Venn
Venn

6. The General Addition Rule
When two events A and B are disjoint, we can use the addition rule for disjoint events from Chapter 14:
P(A or B) =
However, when our events are not disjoint, this earlier addition rule will the probability of both A and B occurring. Thus, we need the General Addition Rule.
P(A) + P(B)
double count

7. The General Addition Rule (cont.)
For any two events A and B,
P(A or B) =
The following Venn diagram shows a situation in which we would use the general addition rule:
P(A) + P(B) – P(A and B)

8. Just Checking
Back in chapter 1 we suggested that you sample some pages of this book at random to see whether they had a graph or other data display. We actually did just that. We drew a representative sample and found the following:
48% of pages had some kind of data display
27% of pages had an equation, and
7% of pages had both
Display these results in a Venn diagram.
What is the probability that a randomly selected sample page had a data display but no equation?
What is the probability that a randomly selected sample page had neither a data display nor an equation? D Eq
.41
.07
.20
.32
.41
.32

9. Assignment
p. 362 #1, 4, 5, 8-12, 14-16

10. Chapter 15 Probability Rules! (2)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

11. It Depends…
Back in Chapter 3, we looked at contingency tables and talked about
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
conditional distributions
P(B|A)
conditional probability

12. 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 the fraction of those outcomes B that also occurred.
Note: P(A) cannot equal 0, since we know that A has occurred.

13. The General Multiplication Rule
When two events A and B are independent, we can use the multiplication rule for independent events from Chapter 14:
P(A and B) =
However, when our events are , this earlier multiplication rule does not work. Thus, we need the General Multiplication Rule.
P(A) x P(B)
not independent

14. 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 General Multiplication Rule:
For any two events A and B,
P(A and B) = or
P(A) x P(B|A)
P(B) x P(A|B)

15. Independence
Independence of two events means that the of one event does not influence the of the other.
With our new notation for conditional probabilities, we can now formalize this definition:
Events A and B are independent whenever (Equivalently, events A and B are independent whenever
outcome
probability
P(B|A) = P(B).
P(A|B) = P(A).)

16. Independent ≠ Disjoint
Disjoint events cannot be independent! Well, why not?
Since we know that disjoint events have no outcomes in common, knowing that one occurred means
Thus, the probability of the second occurring based on our knowledge that the first occurred.
It follows, then, that the two events are
A common error is to treat disjoint events as if they were independent, and apply the for independent events—don’t make that mistake.
the
other didn’t.
changed
not
independent.
Multiplication Rule

17. Depending on Independence
It’s much easier to think about independent events than to deal with
It seems that most people’s natural intuition for probabilities breaks down when it comes to
Don’t fall into this trap: whenever you see probabilities multiplied together, stop and ask whether you think they are really
conditional probabilities
conditional probabilities
independent.

18. Just Checking
Remember our sample of pages in this book from this chapter’s first Just Checking…? We found that
48% of pages had some kind of data display
27% of pages had an equation, and
7% of pages had both
Make a contingency table for the variables display and equation.
What is the probability that a randomly selected sample page with an equation also had a data display?
Are having an equation and having a data display disjoint events?
Are having an equation and having a data display independent events?
D / E
Yes No
Total
0.07
0.48
0.27
1.00
0.41
0.20
0.32
0.52
0.73
P(D|E) =
P(D&E)/P(E) =
0.07/0.27 ≈
0.259
No, pages can have both displays and equations
No, to be independent P(D|E) = P(D), but P(D|E) ≈ and P(D) = 0.48

19. Drawing Without Replacement
Sampling without replacement means that once one individual is drawn it
We often sample without replacement, which doesn’t matter too much when we are dealing with
However, when drawing from a small population, we need to and adjust accordingly.
Drawing without replacement is just another instance of working with
doesn’t go back into the pool.
a large
population.
take note
probabilities
conditional probabilities.

20. Tree Diagrams
A tree diagram helps us think through conditional probabilities by showing sequences of events as paths that look like
Making a tree diagram for situations with probabilities is consistent with our
“ ” mantra.
branches of a tree.
conditional
make a picture

21. Tree Diagrams (cont.)
Figure 15.4 is a nice example of a tree diagram and shows how we multiply the probabilities of the branches together:

22. Assignment
p.362 #2, 3, 6, 7, 13, 18, 19, 24, 25, 27, 32

23. Chapter 15 Probability Rules! (3)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

24. Reversing the Conditioning
Reversing the conditioning of two events is rarely intuitive.
Suppose we want to know P(A|B), but we know only P(A), P(B), and P(B|A).
We also know P(A and B), since
P(A and B) =
From this information, we can find P(A|B):
P(A) x P(B|A)

25. Reversing the Conditioning (cont.)
When we reverse the probability from the conditional probability that you’re originally given, you are actually using
Bayes’s Rule.

26. What Can Go Wrong?
Don’t use a probability rule where a rule is appropriate:
Don’t assume that two events are or without checking that they are.
Don’t find probabilities for samples drawn replacement as if they had been drawn replacement.
Don’t conditioning naively.
Don’t confuse “ ” with “ ”
simple
general
independent
disjoint
without
with
reverse
disjoint
independent

27. Practice
Given a standard deck of cards (52), find the probability of each.
A) If one card is drawn, what is the probability that it is an Ace or Red?
B) Two cards are drawn without replacement. What is the probability they are both Aces?
C) I draw one card and look at it. I tell you it is red. What is the probability it is a heart? Also, what is the probability it is red, given it is a heart?
D) Are a red card and a spade independent? Mutually exclusive?
E) Are a red card and an ace independent? Mutually exclusive?
F) Are a face card and a king independent? Mutually exclusive?
P(ace or red) =
P(ace) + P(red) – P(ace and red) =
4/ /52 – 2/52
≈ 0.538
P(ace and ace) =
P(ace) P(ace|ace) =
4/52 · 3/51 ≈
0.0045
P(heart|red) =
P(heart and red) / P(red) =
13/52 / 26/52 =
0.5
P(red|heart) =
P(heart and red) / P(heart) =
13/52 / 13/52 = 1
P(red|spade) = & P(red) =
26/52 No
Yes
P(red|ace) = & P(red) =
1/2
26/52
Yes No
P(face|king) = & P(face) = 1
12/52 No No

28. Assignment
p. 364 #17, 20, 23, 31, 34, 35, 36, 38, 43, 44