Transcript Document

Chapter 5
Section 3
Independence and
the Multiplication Rule
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 1 of 21
Chapter 5 – Section 3
● Learning objectives
1 Understand independence
2 Use the Multiplication Rule for independent events
3 Compute at-least probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 2 of 21
Chapter 5 – Section 3
● The Addition Rule shows how to compute “or”
probabilities
P(E or F)
under certain conditions
● The Multiplication Rule shows how to compute
“and” probabilities
P(E and F)
also under certain (different) conditions
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 3 of 21
Chapter 5 – Section 3
● Learning objectives
1 Understand independence
2 Use the Multiplication Rule for independent events
3 Compute at-least probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 4 of 21
Chapter 5 – Section 3
● The “disjoint” concept corresponds to “or” and
the Addition Rule … disjoint events and adding
probabilities
● The concept of independence corresponds to
“and” and the Multiplication Rule … independent
events and multiplying probabilities
● Basically, events E and F are independent if they
do not affect each other
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 5 of 21
Chapter 5 – Section 3
● Definition of independence
● Events E and F are independent if the
occurrence of E in a probability experiment does
not affect the probability of event F
● Other ways of saying the same thing
 Knowing E does not give any additional information
about F
 Knowing F does not give any additional information
about E
 E and F are totally unrelated
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 6 of 21
Chapter 5 – Section 3
● Examples of independence
 Flipping a coin and getting a “tail” (event E) and
choosing a card and getting the “seven of clubs”
(event F)
 Choosing one student at random from University A
(event E) and choosing another student at random
from University B (event F)
 Choosing a card and having it be a heart (event E)
and having it be a jack (event F)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 7 of 21
Chapter 5 – Section 3
● If the two events are not independent, then they
are said to be dependent
● Dependent does not mean that they completely
rely on each other … it just means that they are
not independent of each other
● Dependent means that there is some kind of
relationship between E and F – even if it is just a
very small relationship
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 8 of 21
Chapter 5 – Section 3
● Examples of dependence
 Whether Jack has brought an umbrella (event E) and
whether his roommate Joe has brought an umbrella
(event F)
 Choosing a card and having it be a red card (event E)
and having it be a heart (event F)
 The number of people at a party (event E) and the
noise level at the party (event F)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 9 of 21
Chapter 5 – Section 3
● What’s the difference between disjoint events
and independent events?
● Disjoint events can never be independent
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Consider two events E and F that are disjoint
Let’s say that event E has occurred
Then we know that event F cannot have occurred
Knowing information about event E has told us much
information about event F
 Thus E and F are not independent
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 10 of 21
Chapter 5 – Section 3
● Learning objectives
1 Understand independence
2 Use the Multiplication Rule for independent events
3 Compute at-least probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 11 of 21
Chapter 5 – Section 3
● The Multiplication Rule for independent events
states that
P(E and F) = P(E) • P(F)
● Thus we can find P(E and F) if we know P(E)
and P(F)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 12 of 21
Chapter 5 – Section 3
● This is also true for more than two independent
events
● If E, F, G, … are all independent (none of them
have any effects on any other), then
P(E and F and G and …)
= P(E) • P(F) • P(G) • …
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 13 of 21
Chapter 5 – Section 3
● Example
 E is the event “draw a card and get a diamond”
 F is the event “toss a coin and get a head”
 E and F are independent
● P(E and F)
 We first draw a card … with probability 1/4 we get a
diamond
 When we toss a coin, half of the time we will then get
a head, or half of the 1/4 probability, or 1/8 altogether
P(E and F) = 1/8
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 14 of 21
Chapter 5 – Section 3
● Another example

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E is the event “draw a card and get a diamond”
Replace the card into the deck
F is the event “draw a second card and get a spade”
E and F are independent
● P(E and F)
P(E and F) = P(E) • P(F) = 1/4 • 1/4 = 1/16
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 15 of 21
Chapter 5 – Section 3
● The previous example slightly modified

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E is the event “draw a card and get a diamond”
Do not replace the card into the deck
F is the event “draw a second card and get a spade”
E and F are not independent
● Why aren’t E and F independent?
 After we draw a diamond, then 13 out of the
remaining 51 cards are spades … so knowing that we
took a diamond out of the deck changes the
probability for drawing a spade
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 16 of 21
Chapter 5 – Section 3
● Learning objectives
1 Understand independence
2 Use the Multiplication Rule for independent events
3 Compute at-least probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 17 of 21
Chapter 5 – Section 3
● There are probability problems which are stated:
What is the probability that "at least" …
● For example
 At least 1 means 1 or 2 or 3 or 4 or …
 At least 5 means 5 or 6 or 7 or 8 or …
● These calculations can be very long and tedious
 The probability of at least 1 = the probability of 1 + the
probability of 2 + the probability of 3 + the probability
of 4 + …
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 18 of 21
Chapter 5 – Section 3
● There is a much quicker way using the
Complement Rule
● Assume that we are counting something
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E = “at least one” and we wish to compute P(E)
Ec = the complement of E, when E does not happen
Ec = “exactly zero”
Often it is easier to compute P(Ec) first, and then
compute P(E) as 1 – P(Ec)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 19 of 21
Chapter 5 – Section 3
● Example
 We flip a coin 5 times … what is the probability that
we get at least 1 head?
 E = {at least one head}
 Ec = {no heads} = {all tails}
● Ec consists of 5 events … tails on the first flip,
tails on the second flip, … tails on the fifth flip
 These 5 events are independent
 P(Ec) = 1/2 • 1/2 • 1/2 • 1/2 • 1/2 = 1/32
 Thus P(E) = 1 – 1/32 = 31/32
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 20 of 21
Summary: Chapter 5 – Section 3
● The Multiplication Rule applies to independent
events, the probabilities are multiplied to
calculate an “and” probability
● Probabilities obey many different rules
 Probabilities must be between 0 and 1
 The sum of the probabilities for all the outcomes must
be 1
 The Complement Rule
 The Addition Rule (and the General Addition Rule)
 The Multiplication Rule
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 21 of 21
Examples

A manufacturer of exercise equipment knows that
10% of their products are defective. They also
know that only 30% of their customers will
actually use the equipment in the first year after it
is purchased. If there is a one-year warranty on
the equipment, what proportion of the customers
will actually make a valid warranty claim?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 22 of 21
● (3%)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 23 of 21
Examples
● The No Child Left Behind (NCLB) Act of 2001 mandates that by the
year 2014, each public school must achieve 100% proficiency.
(Source: www.ed.gov/nclb) That is, every student must pass a
proficiency exam or the school faces serious sanctions. Suppose that
students in a particular school are all very competent. For each
individual, the probability that they will pass the proficiency exam
(independent of their classmates) is 0.99.
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
a. If there are 500 students in the school, what is the probability that
everyone will pass the exam?
b. What is the probability that at least one of the 500 students will not
pass the proficiency exam (and the school will not meet the
mandate)?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 24 of 21
● (0.00657)
● (0.99343)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 25 of 21
Examples

Repeat problem 1 for a school of 1000 very
competent students (with individual success
probabilities of 0.99).
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 26 of 21
● (0.00004; 0.99996)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 27 of 21
Examples
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In a more typical school, suppose that for each
individual student, the probability that they will
pass the proficiency exam is 0.85.
a. If there are 500 students in the school, what is
the probability that everyone will pass the exam?
Is this event likely?
b. What is the probability that at least one of the
500 students will not pass the proficiency exam
(and the school will not meet the mandate)? Is
this event likely?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 28 of 21
● (5.122×10-36; no)
● (1-5.122×10-36; yes)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 29 of 21
Examples
● The following table gives the number (in millions) of men and women
over the age of 24 at each level of hightest educational attainment.
(Source: U.S. Department of Commerce, Census Bureau, Current
Population Survey, March 2004.)
● a. What is the probability that two randomly selected females over the
age of 24 are both college graduates?

b. What is the probability that two randomly selected people over the
age of 24 are both women?

c. Suppose four people over the age of 24 are randomly selected,
what is the probability that at least one will be a college graduate?

d. Suppose three people over the age of 24 are randomly selected,
what is the probability that at least one will not have attended college?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 30 of 21
●
●
●
●
(0.1250)
(0.2710)
(0.8334)
(0.8500)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 3 – Slide 31 of 21