1. Addition Rule Objectives
Determine if two events are mutually exclusive
Use the Addition Rule to find the probability of two events
Larson/Farber 4th ed

2. Mutually Exclusive Events
Two events A and B cannot occur at the same time A B A B
A and B are mutually
exclusive
A and B are not mutually exclusive
Larson/Farber 4th ed

3. Example: Mutually Exclusive Events
Decide if the events are mutually exclusive. Event A: Roll a 3 on a die. Event B: Roll a 4 on a die.
Solution:
Mutually exclusive (The first event has one outcome, a 3. The second event also has one outcome, a 4. These outcomes cannot occur at the same time.)
Larson/Farber 4th ed

4. Example: Mutually Exclusive Events
Decide if the events are mutually exclusive. Event A: Randomly select a male student. Event B: Randomly select a nursing major.
Solution:
Not mutually exclusive (The student can be a male nursing major.)
Larson/Farber 4th ed

5. The Addition Rule Addition rule for the probability of A or B
The probability that events A or B will occur is
P(A or B) = P(A) + P(B) – P(A and B)
For mutually exclusive events A and B, the rule can be simplified to
P(A or B) = P(A) + P(B)
Can be extended to any number of mutually exclusive events
Larson/Farber 4th ed

6. Example: Using the Addition Rule
You select a card from a standard deck. Find the probability that the card is a 4 or an ace.
Solution:
The events are mutually exclusive (if the card is a 4, it cannot be an ace) 4♣ 4♥ 4♦ 4♠ A♣ A♥ A♦ A♠
44 other cards
Deck of 52 Cards
Larson/Farber 4th ed

7. Example: Using the Addition Rule
You roll a die. Find the probability of rolling a number less than 3 or rolling an odd number.
Solution:
The events are not mutually exclusive (1 is an outcome of both events)
Odd 5 3 1 2 4 6
Less than three
Roll a Die
Larson/Farber 4th ed

8. Solution: Using the Addition Rule
Odd 5 3 1 2 4 6
Less than three
Roll a Die
Larson/Farber 4th ed

9. Example: Using the Addition Rule
The frequency distribution shows the volume of sales (in dollars) and the number of months a sales representative reached each sales level during the past three years. If this sales pattern continues, what is the probability that the sales representative will sell between $75,000 and $124,999 next month?
Sales volume ($)
Months
0–24,999 3
25,000–49,999 5
50,000–74,999 6
75,000–99,999 7
100,000–124,999 9
125,000–149,999 2
150,000–174,999
175,000–199,999 1
Larson/Farber 4th ed

10. Solution: Using the Addition Rule
A = monthly sales between $75,000 and $99,999
B = monthly sales between $100,000 and $124,999
A and B are mutually exclusive
Sales volume ($)
Months
0–24,999 3
25,000–49,999 5
50,000–74,999 6
75,000–99,999 7
100,000–124,999 9
125,000–149,999 2
150,000–174,999
175,000–199,999 1
Larson/Farber 4th ed

11. Example: Using the Addition Rule
A blood bank catalogs the types of blood given by donors during the last five days. A donor is selected at random. Find the probability the donor has type O or type A blood.
Type O
Type A
Type B
Type AB
Total
Rh-Positive
156
139 37 12
344
Rh-Negative 28 25 8 4 65
184
164 45 16
409
Larson/Farber 4th ed

12. Solution: Using the Addition Rule
The events are mutually exclusive (a donor cannot have type O blood and type A blood)
Type O
Type A
Type B
Type AB
Total
Rh-Positive
156
139 37 12
344
Rh-Negative 28 25 8 4 65
184
164 45 16
409
Larson/Farber 4th ed

13. Example: Using the Addition Rule
Find the probability the donor has type B or is Rh-negative.
Type O
Type A
Type B
Type AB
Total
Rh-Positive
156
139 37 12
344
Rh-Negative 28 25 8 4 65
184
164 45 16
409
Solution:
The events are not mutually exclusive (a donor can have type B blood and be Rh-negative)
Larson/Farber 4th ed

14. Solution: Using the Addition Rule
Type O
Type A
Type B
Type AB
Total
Rh-Positive
156
139 37 12
344
Rh-Negative 28 25 8 4 65
184
164 45 16
409
Larson/Farber 4th ed

15. Fundamental Counting Principle
If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m*n.
Can be extended for any number of events occurring in sequence.
Larson/Farber 4th ed

16. Example: Fundamental Counting Principle
You are purchasing a new car. The possible manufacturers, car sizes, and colors are listed. Manufacturer: Ford, GM, Honda Car size: compact, midsize Color: white (W), red (R), black (B), green (G) How many different ways can you select one manufacturer, one car size, and one color? Use a tree diagram to check your result.
Larson/Farber 4th ed

17. Solution: Fundamental Counting Principle
There are three choices of manufacturers, two car sizes, and four colors. Using the Fundamental Counting Principle: 3 ∙ 2 ∙ 4 = 24 ways
Larson/Farber 4th ed

18. Permutations Permutation An ordered arrangement of objects
The number of different permutations of n distinct objects is n! (n factorial)
n! = n∙(n – 1)∙(n – 2)∙(n – 3)∙ ∙ ∙3∙2 ∙1
0! = 1
Examples:
6! = 6∙5∙4∙3∙2∙1 = 720
4! = 4∙3∙2∙1 = 24
Larson/Farber 4th ed

19. Permutations Permutation of n objects taken r at a time
The number of different permutations of n distinct objects taken r at a time ■
where r ≤ n
Larson/Farber 4th ed

20. Example: Finding nPr
Find the number of ways of forming three-digit codes in which no digit is repeated.
Solution:
You need to select 3 digits from a group of 10
n = 10, r = 3
Larson/Farber 4th ed

21. Example: Finding nPr
Forty-three race cars started the 2007 Daytona 500. How many ways can the cars finish first, second, and third?
Solution:
You need to select 3 cars from a group of 43
n = 43, r = 3
Larson/Farber 4th ed

22. Distinguishable Permutations
The number of distinguishable permutations of n objects where n1 are of one type, n2 are of another type, and so on ■
where n1 + n2 + n3 +∙∙∙+ nk = n
Larson/Farber 4th ed

23. Combinations Combination of n objects taken r at a time
A selection of r objects from a group of n objects without regard to order ■
Larson/Farber 4th ed

24. Example: Combinations
A state’s department of transportation plans to develop a new section of interstate highway and receives 16 bids for the project. The state plans to hire four of the bidding companies. How many different combinations of four companies can be selected from the 16 bidding companies?
Solution:
You need to select 4 companies from a group of 16
n = 16, r = 4
Order is not important
Larson/Farber 4th ed

25. Solution: Combinations
Larson/Farber 4th ed

26. 3 Chapter Discrete Probability Distributions
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27. A random variable is a numerical measure of the outcome from a probability experiment, so its value is determined by chance. Random variables are denoted using letters such as X.
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28. A discrete random variable has either a finite or countable number of values. The values of a discrete random variable can be plotted on a number line with space between each point. See the figure.
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29. A continuous random variable has infinitely many values
A continuous random variable has infinitely many values. The values of a continuous random variable can be plotted on a line in an uninterrupted fashion. See the figure.
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30. (b) The number of leaves on a randomly selected Oak tree.
EXAMPLE Distinguishing Between Discrete and Continuous Random Variables
Determine whether the following random variables are discrete or continuous. State possible values for the random variable.
The number of light bulbs that burn out in a room of 10 light bulbs in the next year.
(b) The number of leaves on a randomly selected Oak tree.
(c) The length of time between calls to 911.
Discrete; x = 0, 1, 2, …, 10
Discrete; x = 0, 1, 2, …
Continuous; t > 0
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31. A probability distribution provides the possible values of the random variable X and their corresponding probabilities. A probability distribution can be in the form of a table, graph or mathematical formula.
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32. EXAMPLE A Discrete Probability Distribution
The table to the right shows the probability distribution for the random variable X, where X represents the number of DVDs a person rents from a video store during a single visit. x
P(x)
0.06 1
0.58 2
0.22 3
0.10 4
0.03 5
0.01
© 2010 Pearson Prentice Hall. All rights reserved

33. © 2010 Pearson Prentice Hall. All rights reserved

34. EXAMPLE Identifying Probability Distributions
Is the following a probability distribution? x
P(x)
0.16 1
0.18 2
0.22 3
0.10 4
0.30 5
0.01
© 2010 Pearson Prentice Hall. All rights reserved

35. EXAMPLE Identifying Probability Distributions
Is the following a probability distribution? x
P(x)
0.16 1
0.18 2
0.22 3
0.10 4
0.30 5
-0.01
© 2010 Pearson Prentice Hall. All rights reserved

36. EXAMPLE Identifying Probability Distributions
Is the following a probability distribution? x
P(x)
0.16 1
0.18 2
0.22 3
0.10 4
0.30 5
0.04
© 2010 Pearson Prentice Hall. All rights reserved

37. A probability histogram is a histogram in which the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of that value of the random variable.
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38. EXAMPLE Drawing a Probability Histogram
P(x)
0.06 1
0.58 2
0.22 3
0.10 4
0.03 5
0.01
Draw a probability histogram of the probability distribution to the right, which represents the number of DVDs a person rents from a video store during a single visit.
© 2010 Pearson Prentice Hall. All rights reserved

39. © 2010 Pearson Prentice Hall. All rights reserved

40. EXAMPLE Computing the Mean of a Discrete Random Variable
Compute the mean of the probability distribution to the right, which represents the number of DVDs a person rents from a video store during a single visit. x
P(x)
0.06 1
0.58 2
0.22 3
0.10 4
0.03 5
0.01
© 2010 Pearson Prentice Hall. All rights reserved

41. Because the mean of a random variable represents what we would expect to happen in the long run, it is also called the expected value, E(X), of the random variable.
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42. EXAMPLE Computing the Expected Value of a Discrete Random Variable
A term life insurance policy will pay a beneficiary a certain sum of money upon the death of the policy holder. These policies have premiums that must be paid annually. Suppose a life insurance company sells a $250,000 one year term life insurance policy to a 49-year-old female for $530. According to the National Vital Statistics Report, Vol. 47, No. 28, the probability the female will survive the year is Compute the expected value of this policy to the insurance company. x
P(x)
530
530 – 250,000 = -249,470
Survives
Does not survive
E(X) = 530( ) + (-249,470)( )
= $7.50
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43. © 2010 Pearson Prentice Hall. All rights reserved

44. EXAMPLE. Computing the Variance and Standard Deviation
EXAMPLE Computing the Variance and Standard Deviation of a Discrete Random Variable x
P(x)
0.06 1
0.58 2
0.22 3
0.10 4
0.03 5
0.01
Compute the variance and standard deviation of the following probability distribution which represents the number of DVDs a person rents from a video store during a single visit. x
P(x)
0.06
-1.43
2.0449 1
0.58
-0.91
0.8281 2
0.22
-1.27
1.6129 3
0.1
-1.39
1.9321 4
0.03
-1.46
2.1316 5
0.01
-1.48
2.1904
© 2010 Pearson Prentice Hall. All rights reserved