The Addition Rule and Complements 5.2

● Venn Diagrams provide a useful way to visualize probabilities  The entire rectangle represents the sample space S  The circle represents an event E VENN DIAGRAMS S E

 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 VENN DIAGRAM

● 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 MUTUALLY EXCLUSIVE

● 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) ADDITION RULE

● 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 ADDITION RULE

● 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 EXAMPLE

 The addition rule only applies to events that are disjoint  If the two events are not disjoint, then this rule must be modified WHAT IF NOT DISJOINT?

 The Venn diagram below illustrates how the outcomes {1} and {3} are counted both in event E and event F VENN DIAGRAM

● 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} EXAMPLE

● 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 GENERAL ADDITION RULE

● 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 EXAMPLE

P(E) = P(queen) = 4/52 P(F) = P(heart) = 13/52 P(E and F) = P(queen of hearts) = 1/52, so

 The Probability of an event with the word AND must have that event in ALL of the experiments.  Example: A = 1,3,5,7  B = 2,3,5  So, The outcomes of A and B are 3 and 5 THE WORD AND

 Example  If A = 1,2,3 B = 4, 5 Find P (A and B) Empty Set written { } or Ø EMPTY SET

● 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” COMPLEMENT

● 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 PROBABILITY OF A COMPLEMENT

 The Complement Rule can also be illustrated using a Venn diagram VENN DIAGRAM

 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 SUMMARY