1Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH

EDITIONEDITION

ELEMENTARY STATISTICS Section 3-4 Multiplication Rule: Basics

2Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Finding the Probability of Two or More Selections

Multiple selections

means

Multiplication Rule

3Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

NotationP(A and B) =

P(event A occurs in a first trial and

event B occurs in a second trial)

4Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

P(B A) represents the probability of event B occurring after it is assumed that event A has already occurred (read B A as “B given A”).

Notation for Conditional Probability

5Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Definitions

Independent EventsTwo events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other.

Dependent EventsIf A and B are not independent, they are said to be dependent.

6Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Formal Multiplication Rule

P(A and B) = P(A) • P(B A)

If A and B are independent events, P(B A) is really the same as P(B)

7Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Figure 3-10 Applying the Multiplication Rule

P(A or B)

Multiplication Rule

AreA and B

independent?

P(A and B) = P(A) • P(B A)

P(A and B) = P(A) • P(B)Yes

No

8Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Intuitive Multiplication

When finding the probability that event A occurs in one trial and B occurs in the next trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B takes into account the previous occurrence of event A.

9Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Examples

• A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die.

• Are the independent events? Yes

• Solution:– P(Head and 4) = P(Head) * P(4)

= 1/2 * 1/6 = 1/12

10Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Examples

• A bag contains 3 red balls, 2 blue balls and 5 white balls. Balls are selected with replacement in between.

• Are the independent events? Yes because of the replacement.

• P( Red and Blue)?= P(Red) * P(Blue) = 3/10 * 2/10 = 6/100 = 3/50 =.06

• P( Red and White and Blue)?= P(Red)*P(White)*P(Blue) = 3/10 *5/10* 2/10 = 30/1000 = 3/100 = .03

11Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Examples• A bag contains 3 red balls, 2 blue balls and 5

white balls. Balls are selected without replacement in between.

• Are the independent events? No because there are less balls each time.

• P( Red and White)== P(Red) * P(White|Red) = 3/10 * 5/9 = 15/90 = 1/6 =.167

• P( Red and White and Blue) == P(Red)*P(White|Red)*P(Blue|Red & White) = 3/10 *5/9* 2/8 = 30/720 = 1/24 = .0417

• P( Red and Red)== P(Red and Red) = P(Red)*P(Red|Red) = 3/10 *2/9 = 6/90 = 1/15 = .0667

12Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

STANDARD DECK OF CARDS

• Standard Deck Consists of 52 Cards

• 4 Suits:– Hearts(♥), Clubs(♣), Diamonds(♦), Spades(♠)

• 2 Colors: 26 of each color– Red: Hearts and Diamonds, Black: Clubs and Spades

• 13 Cards in each Suit: – Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen and King

• 3 Face Cards in each Suit – Jack, Queen and King

• 4 of each card– Ex. There are 4 Aces, 4 Kings, etc.

13Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Examples

• Two cards are selected from a deck without replacement in between.

• Are the independent events? No because there are less cards each time.

• P( Heart and Club)== P(Heart) * P(Club|Heart) = 13/52 * 13/51 = 169/2652 =.0637

• P( Ace and Ace)== P(Ace)*P(Ace|Ace) = 4/52 *3/51 = 12/2652 = .00452

14Chapter 3. Section 3-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Small Samples from

Large Populations

If a sample size is no more than 5% of the size of the population, treat the selections as being independent (even if the selections are made without replacement, so they are technically dependent).