1. Independence and the Multiplication Rule
Chapter 5 Section 3
Independence and
the Multiplication Rule

2. Chapter 5 – Section 3 Learning objectives Understand independence
Use the Multiplication Rule for independent events
Compute at-least probabilities 1 2 3

3. 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
The Addition Rule shows how to compute “or” probabilities
P(E or F)‏
under certain conditions

4. Chapter 5 – Section 3 Learning objectives Understand independence
Use the Multiplication Rule for independent events
Compute at-least probabilities 1 2 3

5. 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
The “disjoint” concept corresponds to “or” and the Addition Rule … disjoint events and adding probabilities
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

6. 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
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

7. 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)‏
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)‏
Examples of independence
Flipping a coin and getting a “tail” (event E) and choosing a card and getting the “seven of clubs” (event F)‏
Examples of independence

8. 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
If the two events are not independent, then they are said to be dependent
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

9. Chapter 5 – Section 3 Examples of dependence 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)‏
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)‏
Examples of dependence
Whether Jack has brought an umbrella (event E) and whether his roommate Joe has brought an umbrella (event F)‏
Examples of dependence

10. Chapter 5 – Section 3
What’s the difference between disjoint events and independent events?
Disjoint events can never be independent
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
What’s the difference between disjoint events and independent events?
Disjoint events can never be independent
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
What’s the difference between disjoint events and independent events?
Disjoint events can never be independent
Consider two events E and F that are disjoint
Let’s say that event E has occurred
What’s the difference between disjoint events and independent events?

11. Chapter 5 – Section 3 Learning objectives Understand independence
Use the Multiplication Rule for independent events
Compute at-least probabilities 1 2 3

12. 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)‏

13. 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) • …
This is also true for more than two independent events

14. Chapter 5 – Section 3 Example P(E and F)‏ P(E and F) = 1/8 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
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

15. P(E and F) = P(E) • P(F) = 1/4 • 1/4 = 1/16
Chapter 5 – Section 3
Another example
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
Another example
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

16. Chapter 5 – Section 3 The previous example slightly modified
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
The previous example slightly modified
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

17. Chapter 5 – Section 3 Learning objectives Understand independence
Use the Multiplication Rule for independent events
Compute at-least probabilities 1 2 3

18. 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 + …
There are probability problems which are stated:
What is the probability that "at least" …
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 …

19. Chapter 5 – Section 3
There is a much quicker way using the Complement Rule
Assume that we are counting something
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)‏
There is a much quicker way using the Complement Rule
Assume that we are counting something
E = “at least one” and we wish to compute P(E)‏
Ec = the complement of E, when E does not happen
Ec = “exactly zero”
There is a much quicker way using the Complement Rule
Assume that we are counting something
E = “at least one” and we wish to compute P(E)‏
There is a much quicker way using the Complement Rule
There is a much quicker way using the Complement Rule
Assume that we are counting something
E = “at least one” and we wish to compute P(E)‏
Ec = the complement of E, when E does not happen

20. 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
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}

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

22. 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?

23. (3%)

24. Examples
The No Child Left Behind (NCLB) Act of 2001 mandates that by the year 2014, each public school must achieve 100% proficiency. (Source: 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.
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)?

25. ( )
( )

26. Examples
Repeat problem 1 for a school of 1000 very competent students (with individual success probabilities of 0.99).

27. ( ; )

28. Examples
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?

29. (5.122×10-36; no)
( ×10-36; yes)

30. 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?

31. (0.1250)
(0.2710)
(0.8334)
(0.8500)