(Lesson 11: Multiplication Rule; 4-4) 4.20
Example 4
G.W. Bush won 47.8% of the popular vote in 2000.
(Al Gore won 48.4%, and Ralph Nader won 2.7%.)
Over 105 million voters in the U.S. voted for Presi
(Lesson 11: Multiplication Rule; 4-4) 4.19
PART D: SAMPLING RULE FOR TREATING DEPENDENT EVENTS AS
INDEPENDENT
When we conduct polls, we sample without replacement, so that the same person is
not conta
(Lesson 11: Multiplication Rule; 4-4) 4.18
Solution to Example 3
Because we are drawing cards without replacement, previous draws affect
later draws, and the draws are dependent.
(
P A-1st and K-2nd a
(Lesson 11: Multiplication Rule; 4-4) 4.17
PART C: DEPENDENT EVENTS
()
P B A = the updated probability that B occurs, given that A occurs.
General Multiplication Rule
For events A, B, C, etc.,
(
) ()
(Lesson 11: Multiplication Rule; 4-4) 4.16
Solution to Example 2
Because we are drawing cards with replacement, the draws are independent.
(
)
(
)(
)(
P A-1st and K-2nd and K-3rd = P A-1st P K-2nd P K
(Lesson 11: Multiplication Rule; 4-4) 4.15
The events 3 and Hearts are independent events, because knowing the
rank of a card tells us nothing about its suit, and vice-versa. The occurrence
of one eve
(Lesson 11: Multiplication Rule; 4-4) 4.14
LESSON 11: MULTIPLICATION RULE (SECTION 4-4)
PART A: INDEPENDENT EVENTS
Example 1
Pick (or draw) a card from a standard deck of 52 cards with no Jokers.
(Kno
(Lesson 10: Addition Rule; 4-3) 4.13
Solution to Example 4
We will boldface the entries that correspond to Juniors or students receiving Bs
or Cs. Their sum is 23.
Think About It: Whats an easy way of
(Lesson 10: Addition Rule; 4-3) 4.12
Example 3
Roll two dice.
(
)
16
36
4
=
44.4%
9
P doubles or a "6" on either die =
(
)
A formula would be tricky to apply here. Well use a diagram, instead.
Exampl
(Lesson 10: Addition Rule; 4-3) 4.11
(
)
Subtracting P A and B adjusts for the double-counting from the first two
terms. Consider the following Venn Diagram:
Note: The General Addition Rule works even
(Lesson 10: Addition Rule; 4-3) 4.10
Example 2
Roll one die.
# of elos for which A or B occurs
P or higher than 2 =
even
N
Event A
Event B
=
5
6
Easiest approach: Begin by indicating the appropria
(Lesson 10: Addition Rule; 4-3) 4.09
LESSON 10: ADDITION RULE (SECTION 4-3)
Example 1
Roll one die.
Easiest approach:
4
P or =
even
5
6
Event A
Event B
=
2
3
Here, events A and B are disjoint, or mu
(Lesson 9: Probability Basics; 4-2) 4.08
Approach 3): Subjective Approach
Probabilities here are (hopefully educated) guesstimates.
For example, what is your estimate for:
P a Republican will win the
(Lesson 9: Probability Basics; 4-2) 4.07
Approach 2): Frequentist / Empirical Approach
Let N = total # of trials observed.
If N is large, then:
()
PA
# of trials in which A occurred
N
Example 5
A magi
(Lesson 9: Probability Basics; 4-2) 4.06
Example 4 (Roulette)
18 red slots
18 black slots
2 green slots
38 total
18 9
=
47.4%
38 19
18 9
P black =
=
47.4%
38 19
2
1
P green =
=
5.26%
38 19
()
P red
(Lesson 9: Probability Basics; 4-2) 4.05
We get doubles if both dice yield the same number.
(
)
6
36
1
=
6
P doubles =
Think About It: Assume that the red die is rolled first.
Regardless of the result
(Lesson 9: Probability Basics; 4-2) 4.04
Example 3 (Roll Two Dice 1 red, 1 green)
We roll two standard six-sided dice.
It turns out to be far more convenient if we distinguish between the
dice. For ex
(Lesson 9: Probability Basics; 4-2) 4.03
Example 2 (Roll One Die)
We roll one standard six-sided die.
cfw_
S = 1, 2, 3, 4, 5, 6 .
N = 6 elos.
Notation: We will use shorthand. For example, the event 4
(Lesson 9: Probability Basics; 4-2) 4.02
PART B: ROUNDING
Rounding conventions may be inconsistent.
1
, or rounding off to
3
three significant digits, such as 0.333 (33.3%) or 0.00703 (0.703%). Althou
(Lesson 9: Probability Basics; 4-2) 4.01
CHAPTER 4: PROBABILITY
The French mathematician Laplace once claimed that probability theory is nothing but
common sense reduced to calculation. This is true m
(Lesson 8: Boxplots; also 3-4) 3.34
LESSON 8: BOXPLOTS (ALSO SECTION 3-4)
The Five-Number Summary can be represented graphically as a boxplot, or box-andwhisker plot.
Boxplots can help us compare diff
(Lesson 7: Measures of Relative Standing or Position; 3-4) 3.32
Note: If we assume normality in both classes, we have:
PART B: FRACTILES, OR QUANTILES
We will look at three types of these.
Percentiles
(Lesson 7: Measures of Relative Standing or Position; 3-4) 3.31
z Score Rule of Thumb for Unusual Data Values
Data values whose corresponding z scores are either less than
than 2 (i.e., z < 2 or z > 2
(Lesson 7: Measures of Relative Standing or Position; 3-4) 3.30
LESSON 7: MEASURES OF RELATIVE STANDING OR
POSITION (SECTION 3-4)
How high or low is a data value relative to the others? We want standa
(Lesson 6: Measures of Spread or Variation; 3-3) 3.29
Example 4
We believe that the vast majority of textbooks in a campus bookstore have
prices between $20 and $180. Estimate , the SD of textbook pri