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Addition Rule. Definition. Compound Event Any event combining 2 or more simple events. Notation P ( A or B ) = P (event A occurs or event B occurs or A and B both occur). General Rule for a Compound Event.
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Definition • Compound Event • Any event combining 2 or more simple events Notation P(A or B) = P (event A occurs or event B occurs or A and B both occur)
General Rule for a Compound Event • When finding the probability that event A occurs or event B occurs, find the total number of ways A can occur and the number of ways B can occur, but find the total in such a way that no outcome is counted more than once. • To count each outcome only once, we first need to understand the term mutually exclusive or disjoint.
Mutually Exclusive Events A and B A B A B Two events, A and B, are mutually exclusive, also know as disjoint, if they cannot occur at the same time. A and B are mutually exclusive. A and B are not mutually exclusive.
Mutually Exclusive Events A B 1 2 4 Example: Decide if the two events are mutually exclusive (disjoint). Event A: Roll a number less than 3 on a die. Event B: Roll a 4 on a die. These events cannot happen at the same time, so the events are mutually exclusive or disjoint.
Mutually Exclusive Events A B 2 9 J 10 3 A 7 J J K 4 5 J 8 6 Q Example: Decide if the two events are mutually exclusive (disjoint). Event A: Select a Jack from a deck of cards. Event B: Select a heart from a deck of cards. Because the card can be a Jack and a heart at the same time, the events are not mutually exclusive.
The Addition Rule The probability that event A or B will occur is given by P (A or B) = P (A) + P (B) – P (A and B ). If events A and B are mutually exclusive, then the rule can be simplified to P (A or B) = P (A) + P (B). Example: You roll a die. Find the probability that you roll a number less than 3 or a 4. The events are mutually exclusive. P (roll a number less than 3 or roll a 4) = P (number is less than 3) + P (4)
Applying the Addition Rule Figure 3-6
The Addition Rule Example: A card is randomly selected from a deck of cards. Find the probability that the card is a Jack or the card is a heart. The events are not mutually exclusive because the Jack of hearts can occur in both events. P (select a Jack or select a heart) = P (Jack) + P (heart) – P (Jack of hearts)
The Addition Rule Example: 100 college students were surveyed and asked how many hours a week they spent studying. The results are in the table below. Find the probability that a student spends between 5 and 10 hours or more than 10 hours studying. The events are mutually exclusive. P (5 to10) + P (10) P (5 to10 hours or more than 10 hours) =
Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 45 2223 Example Titanic Passengers • Find the probability of randomly selecting a man or a boy.
Example Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 45 2223 • Find the probability of randomly selecting a man or a boy. • P (man or boy) = * Disjoint (Mutally Exclusive) *
Example Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 45 2223 • Find the probability of randomly selecting a man or • someone who survived.
Example Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 45 2223 • Find the probability of randomly selecting a man or someone who survived. • P(man or survivor) = * NOT Mutally Exclusive (Disjoint)*
Complementary Events All simple events are either in A or A. • A and A • are • mutually exclusive
Rules of Complementary Events P(A) + P(A) = 1 = 1 –P(A) P(A) = 1 –P(A) P(A)
Venn Diagram for the Complement of Event A Figure 3-7