1. CHAPTER 5 Probability: What Are the Chances?
5.2
Probability Rules

2. Probability Rules DESCRIBE a probability model for a chance process.
USE basic probability rules, including the complement rule and the addition rule for mutually exclusive events.
USE a two-way table or Venn diagram to MODEL a chance process and CALCULATE probabilities involving two events.
USE the general addition rule to CALCULATE probabilities.

3. Aim – How can we calculate probability values from events?
H.W. – pg. 314 – 316 #43 – 48, 53, 55

4. Probability Models
In Section 5.1, we used simulation to imitate chance behavior. Fortunately, we don’t have to always rely on simulations to determine the probability of a particular outcome. Descriptions of chance behavior contain two parts:
The sample space S of a chance process is the set of all possible outcomes.
A probability model is a description of some chance process that consists of two parts:
a sample space S and
a probability for each outcome.

5. Example: Building a probability model
Sample Space
36 Outcomes
Since the dice are fair, each outcome is equally likely.
Each outcome has probability 1/36.
Try Exercise 39

6. Example – flipping a coin.
Flip a coin 3 times and write down the probability model for this:
HHH, HHT, HTH, HTT, TTT, TTH, THT, THH
BTW – your graphing calculator can simulate coin flips and other experiments by doing the following:
Going to apps
Press 9
Press 1
Then simulate coin flipping

7. Probability Models
Probability models allow us to find the probability of any collection of outcomes.
An event is any collection of outcomes from some chance process. That is, an event is a subset of the sample space. Events are usually designated by capital letters, like A, B, C, and so on.
If A is any event, we write its probability as P(A).
In the dice-rolling example, suppose we define event A as “sum is 5.”
There are 4 outcomes that result in a sum of 5.
Since each outcome has probability 1/36, P(A) = 4/36.
Suppose event B is defined as “sum is not 5.” What is P(B)?
P(B) = 1 – 4/36 = 32/36

8. Basic Rules of Probability
The probability of any event is a number between 0 and 1.
All possible outcomes together must have probabilities whose sum is exactly 1.
If all outcomes in the sample space are equally likely, the probability that event A occurs can be found using the formula
The probability that an event does not occur is 1 minus the probability that the event does occur.
If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.
Two events A and B are mutually exclusive (disjoint) if they have no outcomes in common and so can never occur together—that is, if P(A and B ) = 0.

9. Basic Rules of Probability
We can summarize the basic probability rules more concisely in symbolic form.
Basic Probability Rules
For any event A, 0 ≤ P(A) ≤ 1.
If S is the sample space in a probability model,
P(S) = 1.
In the case of equally likely outcomes,
Complement rule: P(AC) = 1 – P(A)
Addition rule for mutually exclusive events: If A and B are mutually exclusive,
P(A or B) = P(A) + P(B).

10. A.P. Exam Common Error
When using the complement rule, some students use incorrect notation such as 0.25c = 0.75 or P(A)c = Remember that events have complements, not probabilities. Here is a correct statement:
If P(A) = 0.25, then P(Ac) = 1 – 0.25 = Notice that the c appears in the superscript of event A, not its probability.

11. Example and Practice:
E-Book example: Distance learning.

12. Practice:
Randomly select a student who took the 2013 AP® Statistics exam and record the student’s score. Here is the probability model:
Show that this is a legitimate probability model
Find the probability that the chosen student scored 3 or better.
Score: 1 2 3 4 5
Probability:
0.235
0.188
0.249
0.202
0.126

13. Check our understanding:
Choose an American adult at random. Define two events:
A = the person has a cholesterol of 240 milligrams per deciliter of blood or above (high cholesterol).
B = The person has a cholesterol of 200 to 239 milligrams per deciliter (borderline high cholesterol).
According to the American Heart Association, P(A) = 0.16, P(B) = 0.29.
Explain why events A and B are mutually exclusive.
Solution:
A person can’t have both high and borderline cholesterol.

14. Check our understanding continued:
Find P(A or B).
Solution:
P(A or B) = P(A) + P(B) = = 0.45.
If C is the event that the person chosen has normal cholesterol (below 200 milligrams), what is P(C)?
P(C) = 1 – 0.45 = 0.55

15. Aim – How can we evaluate two way tables within the context of probability measures?
H.W. pg. 316 – 317 #49, 50, 53, 55bcd

16. Two-Way Tables and Probability
When finding probabilities involving two events, a two-way table can display the sample space in a way that makes probability calculations easier.
Consider the example on page 309. Suppose we choose a student at random. Find the probability that the student
has pierced ears.
is a male with pierced ears.
is a male or has pierced ears.
Try Exercise 39
Define events A: is male and B: has pierced ears.
(a) Each student is equally likely to be chosen. 103 students have pierced ears. So, P(pierced ears) = P(B) = 103/178.
(b) We want to find P(male and pierced ears), that is, P(A and B). Look at the intersection of the “Male” row and “Yes” column. There are 19 males with pierced ears. So, P(A and B) = 19/178.
(c) We want to find P(male or pierced ears), that is, P(A or B). There are 90 males in the class and 103 individuals with pierced ears. However, 19 males have pierced ears – don’t count them twice!
P(A or B) = ( )/178. So, P(A or B) = 174/178

17. General Addition Rule for Two Events
We can’t use the addition rule for mutually exclusive events unless the events have no outcomes in common.
Try Exercise 39
If A and B are any two events resulting from some chance process, then
P(A or B) = P(A) + P(B) – P(A and B)
General Addition Rule for Two Events

18. Check understanding:
Standard deck of playing cards (with jokers removed) consists of 52 cards in 4 suits – clubs, diamonds, spades, and hearts. Each suit has 13 cards, with denominations of 2 – 10, jack, queen, king, and ace. The face cards are the jack, queen, and king. If we shuffle the deck thoroughly and deal one card. Define events as follows:
F = getting a face card.
H = getting a heart.
Make a two-way table that displays the sample space.
Find P(F and H).
Explain why P(F or H) ≠ P(F) + P(H).
Find P(F or H)

19. Venn Diagrams and Probability
Because Venn diagrams have uses in other branches of mathematics, some standard vocabulary and notation have been developed.
The complement AC contains exactly the outcomes that are not in A.
The events A and B are mutually exclusive (disjoint) because they do not overlap. That is, they have no outcomes in common.
Try Exercise 39

20. Venn Diagrams and Probability
The intersection of events A and B (A ∩ B) is the set of all outcomes in both events A and B.
The union of events A and B (A ∪ B) is the set of all outcomes in either event A or B.
Try Exercise 39
Hint: To keep the symbols straight, remember ∪ for union and ∩ for intersection.

21. Venn Diagrams and Probability
Recall the example on gender and pierced ears. We can use a Venn diagram to display the information and determine probabilities.
Define events A: is male and B: has pierced ears.
Try Exercise 39

22. Practice – handout:
Practice textbook and handout

23. Probability Rules DESCRIBE a probability model for a chance process.
USE basic probability rules, including the complement rule and the addition rule for mutually exclusive events.
USE a two-way table or Venn diagram to MODEL a chance process and CALCULATE probabilities involving two events.
USE the general addition rule to CALCULATE probabilities.