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. 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.

4. Example: Flipping a Coin
Probability Model:
Outcome
Heads
Tails
Probability

5. Example
Roll one fair six-sided die. Give a probability model for this chance process.

6. 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).
Examples of events when rolling a die:
Landing on 3
Landing on a number less than 4
Landing on an even number

7. Complement
For any event A, the complement of A includes all the rest of the outcomes in the sample space.
We refer to the complement of an event A as “not A” and denote it by AC.
Example: Name the complement when rolling a die.
Landing on 3
Landing on a number less than 4
Landing on an even number

8. Mutually Exclusive
Examples: Landing on heads and tails
Rolling a 3 and Rolling a number larger than 4
Drawing a diamond and drawing a spade
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. Example: Building a probability model
Example: Find the probability that the sum is 4.
Find the probability that the sum is not 11.
Try Exercise 39

10. 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.

11. 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).

12. Example
Randomly select a student who took the 2015 AP Statistics exam and record the student’s score. Here it the probability model.
Show that this is a legitimate probability model.
Find the probability that the chosen student scored 3 or better.
Find the probability that the chosen student didn’t get a 1.

13. Example
Imagine spinning this spinner three times. Write the probability model for the number of times the spinner lands on the black portion of the circle. Then use it to find the probability of stopping on the black portion at least 1 time in three spins.

14. 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.

15. Homework
Pg #39-47 all

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. Example
The 120 students who took AP Stats at a local high school one year could pay for UMSL credit for the class (or not) and could sign up to take the AP exam to try to earn college credit that way (or not). Overall, 32 students signed up for UMSL credit, 56 took the AP exam, and 8 students both signed up for UMSL credit and took the AP exam.
a. Complete the two-way table that displays the data.
Suppose we choose a student from among those
AP Stats classes at random. Find the probability
that the student:
b. signed up for UMSL credit
c. signed up for UMSL credit and took the AP exam.
d. signed up for UMSL credit or took the AP exam.

18. P(A or B) = P(A) + P(B) – P(A and B)
General Addition Rule for Two Events
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)
Note: This formula is on the formula sheet!! What happens if the events are mutually exclusive?

19. Example
A standard deck of playing cards (with jokers removed) consists of 52 cards in four suits-clubs, diamonds, hearts, and spades. Each suit has 13 cards, with denominations ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. The jack, queen, and king are referred to as “face cards.” Imagine that we shuffle the deck thoroughly and deal one card. Let’s define events A: getting a face card and B: getting a heart. 1. Make a two-way table that displays the sample space. 2. Find P(A and B). 3. Explain why P(A or B) does not equal P(A) +P(B). Then use the general addition rule to find P(A or B).

20. 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

21. 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.

22. 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

23. Example
At one point, among the movie princesses from a particular studio, 25% had blonde hair, 25% had blue eyes, and 10% had both. Suppose we randomly select a princess from this studio at the time in question. c. Find the probability that the princess has blonde hair or blue eyes. d. Find the probability that the princess has neither blonde hair nor blue eyes. e. Find the probability the princess has blonde hair only.

24. Example

25. 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.