1

Chapter 6: Probability—The Study of Randomness

6.1 The Idea of Probability

6.2 Probability Models

6.3 General Probability Rules

2

Simple Question:

• If tossing a coin, what is the probability of the coin turning up heads?

• Most of you probably answered 50%, but how do you know this to be so?

3

Probability

• Probability is the branch of mathematics that describes the pattern of chance outcomes.

• The heart of statistics, and thus the heart of this course, is in statistical inference.– Probability calculations are the basis for inference.

• Mathematical probability is an idealization based on imagining what would happen in an indefinitely long series of trials.

4

Random Behavior

• Many observable phenomena are random: the relative frequencies of outcomes seem to settle down over the long haul.

• The big idea of probability: chance (“random”) behavior is unpredictable in the short run, but has regular and predictable patterns in the long run.

• See Example 6.1, p. 331

5

Figure 6.1, p. 331

6

Simulation Problem

• Problem 6.3, p. 334

7

Probability Definitions

• Sample Space (S)—set of all possible outcomes.– See Example 6.3, p. 336 (rolling two dice)

• Event—a particular outcome or set of outcomes; a subset of the sample space.

• Probability—the number of times an event can occur within the sample space divided by the sample space.– What is the probability of getting a sum ≤5

when rolling two dice?

8

Figure 6.2, p. 336

9

Multiplication Principle

• Multiplication (counting) principle– If you can do one task in a ways and a second task in b

ways, then both tasks can be done in a x b ways.– How many outcomes are possible if we flip 4 coins?– Apply the multiplication rule to the 2-dice problem.

• Example 6.5, p. 337– Tree diagram

10

Figure 6.3, p. 338

11

HW Exercises

• 6.11, p. 340• 6.14 a, b p. 341• 6.18, p. 342• Reading, pp. 335-352

12

Probability Rules (Box on p. 343)

• Rule 1. If P(A) is the probability of an event A, then:

1)(0 AP

• Rule 2. If S is the sample space in a probability model, then:

1)( SP

• Rule 3. The complement (Ac) of an event A is the event that A does not occur. The complement rule states:

)(1)( APAP c

13

Rule 4. Two events are disjoint (mutually exclusive) ifthey have no outcomes in common and can never occursimultaneously. If A and B are disjoint, then:

Rule 5. Two events A and B are independent if knowingthat one occurs does not change the probability that the other occurs. If A and B are independent, then:

)()() and ( BPAPBAP

Probability Rules, cont.(Box on p. 343 and p. 351)

14

Independence Example

• What is the probability of drawing two hearts on successive draws from a standard deck of playing cards?– With replacement– Without replacement

15

Set Notation

B.union A read, ... Bor A BA

BABA intersect

set)(empty :events exclusivemutually For BA

16

In-Class Problems

• 6.19, p. 348

• Look at example 6.10, p. 345– 6.26, p. 349

• More Problems:• 6.27, 6.28, p. 354• 6.31, 6.32, 6.33, p. 355

17

HW

• Problems:– 6.36, 6.37, p. 357– 6.42, p. 359

• Read through p. 370.

18

More on Independence and Disjoint (Mutually Exclusive) Events

• If I roll two dice, then each die can roll a 3. Neither die influences the other, so they are independent. But since I can roll a 3 on each die simultaneously, they are not disjoint events.

+++++++++++++++++++++++++++++++++++++++• A bag contains 3 red balls and 2 green balls. A ball is drawn from

the bag, its color is noted, and the ball is set aside. Then a second ball is drawn and its color is noted.– Let event A be the event that the first ball is red. Let event B be

the event that the second ball is red.• Events A and B are not disjoint because both balls can be red.

Events A and B are not independent because whether the first ball is red or not alters the probability of the second ball being red.

+++++++++++++++++++++++++++++++++++++++• Two events that are disjoint cannot be independent.

– See last full paragraph on p. 352.

19

6.3 General Probability Models

20

General Addition Rule forUnions of Two Events

• Rule 4 (Addition Rule). Two events are disjoint (mutually exclusive) if they have no outcomes in common and can never occur simultaneously. If A and B are disjoint, then:

)()() ( BPAPBorAP

• The above rule does not work if two events are not disjoint (that is, they are not mutually exclusive). Here is the general rule for addition for unions of two events:

) and ()()()or ( BAPBPAPBAP

21

Venn Diagrams

• Venn Diagrams can be very useful in helping find probabilities of unions.

22

Practice Problems

• 6.46, p. 364• 6.47, p. 364• 6.52, p. 365• 6.51, p. 365

More Practice

• pp. 364-365: 6.48, 6.49, 6.50, 6.53

23

24

Conditional Probability

• Probability of an event given, or under the condition that, we know another event.– See example 6.18, p. 366

• Notation:

)/( )/( ABPorBAP

25

General Multiplication Rulefor any Two Events

)(

) and ()/(

AP

BAPABP

)/()() and ( ABPAPBAP

Note: If events A and B are independent, then: )()/( BPABP

26

Practice Problems

• 6.54 and 6.55, pp. 369-370

27

HW

• Problems, pp. 370-371:– 6.56, 6.58, 6.60– 6.61 (include a Venn diagram for part b)

28

Tree Diagrams

• See Examples 6.22-6.23, pp. 372-373

• The probability of reaching the end of any complete branch is the product of the probabilities written on each segment.– Probabilities after the first level of segments are

conditional probabilities.

29

Example 6.23, p. 372

30

Example

• The weather forecaster at the radio station reports that if it rains on a given day in November, the probability of rain the next day is 70%. If it does not rain on a given day, then the probability of rain the next day is 40%. What is the probability that it will rain on Friday if it is raining on Wednesday?– Draw a tree diagram to help solve this problem.

31

Practice Problems

• 6.70, p. 381– Use a tree diagram to answer (b).– Then, use the general multiplication rule to

confirm your answer.

• 6.62 p. 377• 6.67, 6.69 p. 380• 6.63, p. 378• 6.64, p. 378

– Use a tree diagram.

• 6.73, 6.74 p. 381

32

Homework

• 6.63, p. 378• 6.64, p. 378

– Use a tree diagram.

• 6.73, 6.74 p. 381

33

Additional Practice Problems

• Problems, pp. 383-385:– 6.78, 6.79, 6.82, 6.83, 6.84, 6.86