Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 1 of 21 Chapter 5 Section 2 The Addition Rule and Complements

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 2 of 21 Chapter 5 – Section 2 ●Learning objectives  Use the Addition Rule for disjoint events  Use the General Addition Rule  Compute the probability of an event using the Complement Rule 1 2 3

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 3 of 21 Chapter 5 – Section 2 ●Learning objectives  Use the Addition Rule for disjoint events  Use the General Addition Rule  Compute the probability of an event using the Complement Rule 1 2 3

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 4 of 21 Chapter 5 – Section 2 ●Venn Diagrams provide a useful way to visualize probabilities  The entire rectangle represents the sample space S S ●Venn Diagrams provide a useful way to visualize probabilities  The entire rectangle represents the sample space S  The circle represents an event E E

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 5 of 21 Chapter 5 – Section 2 ●In the Venn diagram below  The sample space is {0, 1, 2, 3, …, 9}  The event E is {0, 1, 2}  The event F is {8, 9}  The outcomes {3}, {4}, {5}, {6}, {7} are in neither event E nor event F

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 6 of 21 Chapter 5 – Section 2 ●Two events are disjoint if they do not have any outcomes in common ●Another name for this is mutually exclusive ●Two events are disjoint if they do not have any outcomes in common ●Another name for this is mutually exclusive ●Two events are disjoint if it is impossible for both to happen at the same time ●Two events are disjoint if they do not have any outcomes in common ●Another name for this is mutually exclusive ●Two events are disjoint if it is impossible for both to happen at the same time ●E and F below are disjoint

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 7 of 21 Chapter 5 – Section 2 ●For disjoint events, the outcomes of (E or F) can be listed as the outcomes of E followed by the outcomes of F ●There are no duplicates in this list ●For disjoint events, the outcomes of (E or F) can be listed as the outcomes of E followed by the outcomes of F ●There are no duplicates in this list ●The Addition Rule for disjoint events is P(E or F) = P(E) + P(F)‏ ●For disjoint events, the outcomes of (E or F) can be listed as the outcomes of E followed by the outcomes of F ●There are no duplicates in this list ●The Addition Rule for disjoint events is P(E or F) = P(E) + P(F)‏ ●Thus we can find P(E or F) if we know both P(E) and P(F)‏

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 8 of 21 Chapter 5 – Section 2 ●This is also true for more than two disjoint events ●If E, F, G, … are all disjoint (none of them have any outcomes in common), then P(E or F or G or …) = P(E) + P(F) + P(G) + … ●This is also true for more than two disjoint events ●If E, F, G, … are all disjoint (none of them have any outcomes in common), then P(E or F or G or …) = P(E) + P(F) + P(G) + … ●The Venn diagram below is an example of this

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 9 of 21 Chapter 5 – Section 2 ●Example ●In rolling a fair die, what is the chance of rolling a {2 or lower} or a {6}? ●Example ●In rolling a fair die, what is the chance of rolling a {2 or lower} or a {6}?  The probability of {2 or lower} is 2/6 ●Example ●In rolling a fair die, what is the chance of rolling a {2 or lower} or a {6}?  The probability of {2 or lower} is 2/6  The probability of {6} is 1/6 ●Example ●In rolling a fair die, what is the chance of rolling a {2 or lower} or a {6}?  The probability of {2 or lower} is 2/6  The probability of {6} is 1/6  The two events {1, 2} and {6} are disjoint ●Example ●In rolling a fair die, what is the chance of rolling a {2 or lower} or a {6}  The probability of {2 or lower} is 2/6  The probability of {6} is 1/6  The two events {1, 2} and {6} are disjoint ●The total probability is 2/6 + 1/6 = 3/6 = 1/2

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 10 of 21 Chapter 5 – Section 2 ●Learning objectives  Use the Addition Rule for disjoint events  Use the General Addition Rule  Compute the probability of an event using the Complement Rule 1 2 3

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 11 of 21 Chapter 5 – Section 2 ●The addition rule only applies to events that are disjoint ●If the two events are not disjoint, then this rule must be modified ●Some outcomes will be double counted

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 12 of 21 Chapter 5 – Section 2 ●The Venn diagram below illustrates how the outcomes {1} and {3} are counted both in event E and event F The overlapping region is (E and F)‏

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 13 of 21 Chapter 5 – Section 2 ●Example ●In rolling a fair die, what is the chance of rolling a {2 or lower} or an even number? ●Example ●In rolling a fair die, what is the chance of rolling a {2 or lower} or an even number?  The probability of {2 or lower} is 2/6 ●Example ●In rolling a fair die, what is the chance of rolling a {2 or lower} or an even number?  The probability of {2 or lower} is 2/6  The probability of {2, 4, 6} is 3/6 ●Example ●In rolling a fair die, what is the chance of rolling a {2 or lower} or an even number?  The probability of {2 or lower} is 2/6  The probability of {2, 4, 6} is 3/6  The two events {1, 2} and {2, 4, 6} are not disjoint ●Example ●In rolling a fair die, what is the chance of rolling a {2 or lower} or an even number?  The probability of {2 or lower} is 2/6  The probability of {2, 4, 6} is 3/6  The two events {1, 2} and {2, 4, 6} are not disjoint  The total probability is not 2/6 + 3/6 = 5/6 ●Example ●In rolling a fair die, what is the chance of rolling a {2 or lower} or an even number?  The probability of {2 or lower} is 2/6  The probability of {2, 4, 6} is 3/6  The two events {1, 2} and {2, 4, 6} are not disjoint  The total probability is not 2/6 + 3/6 = 5/6  The total probability is 4/6 because the event is {1, 2, 4, 6}

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 14 of 21 Chapter 5 – Section 2 ●For the formula P(E) + P(F), all the outcomes that are in both events are counted twice ●Thus, to compute P(E or F), these outcomes must be subtracted (once)‏ ●For the formula P(E) + P(F), all the outcomes that are in both events are counted twice ●Thus, to compute P(E or F), these outcomes must be subtracted (once)‏ ●The General Addition Rule is P(E or F) = P(E) + P(F) – P(E and F)‏ ●This rule is true both for disjoint events and for not disjoint events

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 15 of 21 Chapter 5 – Section 2 ●Example ●When choosing a card at random out of a deck of 52 cards, what is the probability of choosing a queen or a heart?  E = “choosing a queen”  F = “choosing a heart” ●Example ●When choosing a card at random out of a deck of 52 cards, what is the probability of choosing a queen or a heart?  E = “choosing a queen”  F = “choosing a heart” ●E and F are not disjoint (it is possible to choose the queen of hearts), so we must use the General Addition Rule

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 16 of 21 Chapter 5 – Section 2 ●P(E) = P(queen) = 4/52 ●P(F) = P(heart) = 13/52 ●P(E and F) = P(queen of hearts) = 1/52, so

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 17 of 21 Chapter 5 – Section 2 ●Learning objectives  Use the Addition Rule for disjoint events  Use the General Addition Rule  Compute the probability of an event using the Complement Rule 1 2 3

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 18 of 21 Chapter 5 – Section 2 ●The complement of the event E, written E c, consists of all the outcomes that are not in that event ●Examples  Flipping a coin … E = “heads” … E c = “tails” ●The complement of the event E, written E c, consists of all the outcomes that are not in that event ●Examples  Flipping a coin … E = “heads” … E c = “tails”  Rolling a die … E = {even numbers} … E c = {odd numbers} ●The complement of the event E, written E c, consists of all the outcomes that are not in that event ●Examples  Flipping a coin … E = “heads” … E c = “tails”  Rolling a die … E = {even numbers} … E c = {odd numbers}  Weather … E = “will rain” … E c = “won’t rain”

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 19 of 21 Chapter 5 – Section 2 ●The probability of the complement E c is 1 minus the probability of E ●This can be shown in one of two ways  It’s obvious … if there is a 30% chance of rain, then there is a 70% chance of no rain ●The probability of the complement E c is 1 minus the probability of E ●This can be shown in one of two ways  It’s obvious … if there is a 30% chance of rain, then there is a 70% chance of no rain  E and E c are two disjoint events that add up to the entire sample space

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 20 of 21 Chapter 5 – Section 2 ●The Complement Rule can also be illustrated using a Venn diagram Entire region The area of the region outside the circle represents E c

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 21 of 21 Summary: Chapter 5 – Section 2 ●Probabilities obey additional rules ●For disjoint events, the Addition Rule is used for calculating “or” probabilities ●For events that are not disjoint, the Addition Rule is not valid … instead the General Addition Rule is used for calculating “or” probabilities ●The Complement Rule is used for calculating “not” probabilities

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 22 of 21 Examples According to the U.S. Census Bureau, among males over the age of 24, 15% did not complete high school, 31% completed high school, 17% attended some college without graduating, and 37% are college graduates. (Source: U.S. Department of Commerce, Census Bureau, Current Population Survey, March 2004.) Compute the probability that a randomly selected male in the United States who is over the age of 24 will not have attended college.

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 23 of 21 Examples ●Let E be the event that a randomly selected person in the U.S. who is over the age of 24 has attended college. Let F be the event that a randomly selected U.S. resident over the age of 24 is female. According to the U.S. Census Bureau, P(E) = 0.47 and P(F) = 0.52, and P(E and F) is (Source: U.S. Department of Commerce, Census Bureau, Current Population Survey, March 2004.) a. What is the probability that a randomly selected person in the U.S. over the age of 24 is a male? b. What is the probability that a randomly selected person in the U.S. over the age of 24 is a female or has attended college? c. What is the probability that a randomly selected person in the U.S. over the age of 24 is a male and has not attended college? d. What is the probability that a randomly selected person in the U.S. over the age of 24 is a female and has not attended college?

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 2 – Slide 24 of 21 ●(0.48) ●(0.72) ●(0.28) ●(0.25)