(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 President in 2000.
Of those, three are randomly selected wit
(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 contacted twice. Technically, the selections are dependent.
(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 and K-3rd
)(
(
)
)(
= P A-1st P K-2nd A-1st P K-3rd A-1s
(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.,
(
) () ( )
P ( A and B and C ) = P ( A) P ( B A) P ( C A and B
(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-3rd
111
13 13 13
1
=
0.000455
2197
)
=
(
)
Tree
PART
(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 event does not change our probability assessment for the o
(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.
(Know this setup!)
()
P3=
(
4
1
=
52 13
)
P hearts =
(
(The
(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 determining that the sum is 23, aside from
adding the
(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.
Example 4
All 26 students in a class are passing, but they se
(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 if A and B are mees.
In that special case, P A and B =
(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 appropriate elos.
Here, events A and B are not mees. Both events
(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 mutually exclusive events (well say mees)
in that they ca
(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 next U.S. Presidential election ?
(
)
An adjustable col
(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 magicians coin comes up heads (H) 255 times and tails (T) 2
(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 =
(
)
(
)
(
)
(
)
(
)
The casino pays even money for r
(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 on the red die, what is
P g matches r ?
(
(
)
)
4
36
1
(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 example, we can color one die red and the other die green
(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 means
the die comes up 4 when it is rolled. It is a si
(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%). Although
there is some controversy about the use of percents
(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 many times, but not always .
LESSON 9: PROBABILITY BASIC
(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 different populations, such as men and women.
See the last
(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
Example
Because 92% of the scores lie below 85 points,
(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 ) are often considered unusual.
2 or greater
In other
(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 standardized measures
that will work for practically all popu
(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 prices in the
bookstore.
Solution to Example 4
Realistic r