2. Chapter 15
Probability Rules!
Copyright © 2010 Pearson Education, Inc.

3. 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  B) = P(A) + P(B)
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 General Addition Rule.
Let’s look at a picture…

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

5. The General Addition Rule (cont.)
A survey of college students found that 56% live on campus, 62% are on a campus meal program, and 42% do both.
What is the probability that:
a student either lives or eats on campus?
a student lives off campus and doesn’t have a meal program?
a student lives in a dorm but doesn’t have a meal program?

6. It Depends…
Back in Chapter 3, we looked at contingency tables and talked about conditional distributions.
When we want the probability of an event from a conditional distribution, we write P(B|A) and pronounce it “the probability of B given A.”
A probability that takes into account a given condition is called a conditional probability.

7. 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.

8. It Depends… (cont.)
Our survey found that 56% live on campus, 62% have a meal program, 42% do both.
While dining in a campus facility open only to students with meal programs you meet someone new. What is the probability they live on campus?

9. 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  B) = P(A) x P(B)
However, when our events are not independent, this earlier multiplication rule does not work. Thus, we need the General Multiplication Rule.

10. 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  B) = P(A)  P(B|A) or
P(A  B) = P(B)  P(A|B)

11. 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 independent whenever P(B|A) = P(B). (Equivalently, events A and B are independent whenever P(A|B) = P(A).)

12. Independence
Events A and B are independent whenever P(B|A) = P(B).
Are living on campus and having a meal plan independent? Are they disjoint?

13. 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 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, and apply the Multiplication Rule for independent events—don’t make that mistake.

14. 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.

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

16. Practice with probabilities
Consider drawing cards from a standard deck of cards, with jokers removed.
One card is drawn. What is the probability it is an ace or a red card?
Two cards are drawn without replacement. What is the probability they are both aces?
If cards are drawn without replacement, what is the probability I get 5 hearts in a row?
I draw a card and tell you it is red. What is the probability it is a heart?
I draw a card and tell you it is a heart. What is the probability it is red?

17. Are “red card” and “spade” independent? Mutually exclusive?
Are “red card” and “ace” independent? Mutually exclusive?
Are “face card” and “king” independent? Mutually exclusive?

18. Checking for Independence
Students in grades 4, 5, and 6 were surveyed as to what their primary goal was.
If we select a student at random, what is
P(girl)
P(girl ∩ popular)
P(grades|boy)
Grades
Popular
Sports
total
Boy
117 50 60
227
Girl
130 91 30
251
Total
247
141 90
478

19. P(sports|girl)
P(sports|boy)
P(sports)
If events are independent, the conditional probability should be the same as the overall probability.

20. 28 Death penalty Favor Oppose Republican 0.26 0.04 Democrat 0.12 0.24
Other
0.10 28

21. #38
23% of adults smoke cigarettes.
57% of smokers and 13% of non-smokers develop a certain lung condition by age 60.
How do these statistics indicate that lung condition and smoking are not independent?
What is the probability that a randomly selected 60 year old has this lung condition?
#40 What is the probability that someone with the condition was a smoker?

22. Building a contingency table
Drivers suspected of drunk driving are given tests to determine blood alcohol content. 78% of suspects get a breath test, 36% a blood test, and 22% get both.
Are any of these numbers marginal probabilities?
Are any joint probabilities?
Breath test
No breath test
total
Blood test
No blood test

23. Building a contingency table
Are giving a blood test and a breath test mutually exclusive?
Are giving the two tests independent?
Breath test
No breath test
total
Blood test
No blood test

24. In our text book, 48% of pages have a data display, 27% have an equation, 7% have both.
Make a contingency table for the variables display and equation.
What is the probability for a randomly selected page with an equation to also have a data display?
Are these disjoint events?
Are these independent events?

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

26. Tree Diagrams (cont.)
Figure 15.5 is a nice example of a tree diagram and shows how we multiply the probabilities of the branches together.
All the final outcomes are disjoint and must add up to one.
We can add the final probabilities to find probabilities of compound events.

27. Making a tree diagram
A recent study found that in 77% of all accidents the driver was wearing a seatbelt. 92% of those drivers escaped serious injury, while only 63% of those not wearing seatbelts escaped injury.
Construct a tree diagram.
What is the probability that a driver who was seriously injured wasn’t wearing a seatbelt?

28. What Can Go Wrong?
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 find probabilities for samples drawn without replacement as if they had been drawn with replacement.
Don’t reverse conditioning naively.
Don’t confuse “disjoint” with “independent.”

29. What have we learned?
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 and that reversing the conditioning can give surprising results.

30. What have we learned? (cont.)
Venn diagrams, tables, and tree diagrams help organize our thinking about probabilities.
We now know more about independence—a sound understanding of independence will be important throughout the rest of this course.