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5.2

5.2. The Addition Rule and Complements. Venn Diagrams. Venn Diagrams provide a useful way to visualize probabilities The entire rectangle represents the sample space S The circle represents an event E. S. E. Venn Diagram. In the Venn diagram below The sample space is {0, 1, 2, 3, …, 9}

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5.2

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  1. 5.2 The Addition Rule and Complements

  2. Venn Diagrams • Venn Diagrams provide a useful way to visualize probabilities • The entire rectangle represents the sample space S • The circle represents an event E S E

  3. Venn Diagram • 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

  4. Mutually Exclusive • Two events are disjoint if they do not have any outcomes in common • Another name for this is mutuallyexclusive • Two events are disjoint if it is impossible for both to happen at the same time • E and F below are disjoint

  5. Addition Rule • 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 AdditionRule 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)

  6. 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

  7. 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

  8. What if not disjoint? • The addition rule only applies to events that are disjoint • If the two events are not disjoint, then this rule must be modified

  9. Venn Diagram • The Venn diagram below illustrates how the outcomes {1} and {3} are counted both in event E and event F

  10. 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}

  11. General Addition Rule • 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 GeneralAdditionRule 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

  12. 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

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

  14. The Word AND • 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

  15. Empty Set • Example • If A = 1,2,3 B = 4, 5 Find P (A and B) Empty Set written { } or Ø

  16. Complement • The complement of the event E, written Ec, consists of all the outcomes that are not in that event • Examples • Flipping a coin … E = “heads” … Ec = “tails” • Rolling a die … E = {even numbers} … Ec = {odd numbers} • Weather … E = “will rain” … Ec = “won’t rain”

  17. Probability of a Complement • The probability of the complement Ec 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 Ec are two disjoint events that add up to the entire sample space

  18. Venn Diagram • The Complement Rule can also be illustrated using a Venn diagram

  19. Summary • 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

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