1 chapter 3. section 3-4. triola, elementary statistics, eighth edition. copyright 2001. addison...
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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).
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