Section 4-3 The Addition Rule

COMPOUND EVENT A compound event is any event combining two or more simple events. NOTATION P(A or B) = P(in a single trial, event A occurs or event B occurs or they both occur)

GENERAL RULE FOR FINDING THE PROBABILITY OF 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.

FORMAL ADDITION RULE

INTUITIVE ADDITION RULE To find P(A or B), find the sum of the number of ways event A can occur and the number of ways event B can occur, adding in such a way that every outcome is counted only once. P(A or B) is equal to that sum, divided by the total number of outcomes in the sample space.

DISJOINT EVENTS Events A and B are disjoint (or mutually exclusive) if they cannot both occur together.

OBSERVATIONS ON DISJOINT EVENTS If two events, A and B, are disjoint, then P(A and B) = 0. If events A and B are disjoint, then P(A or B) = P(A) + P(B).

APPLYING THE ADDITION RULE P(A or B) Addition Rule Are A and B disjoint ? P(A or B) = P(A) + P(B) − P(A and B) P(A or B) = P(A) + P(B) Yes No Disjoint events cannot happen at the same time. They are separate, nonoverlapping events.

EXAMPLE The data in the chart below represent the marital status of males and females 18 years or older in the US in Use it to answer the questions on the next slide. (Source: US Census Bureau) Males (in millions) Females (in millions) Totals (in millions) Never Married Married Widowed Divorced Totals (in millions)

EXAMPLE (CONCLUDED) 1.Determine the probability that a randomly selected United States resident 18 years or older is male. 2.Determine the probability that a randomly selected United States resident 18 years or older is widowed. 3.Determine the probability that a randomly selected United States resident 18 years or older is widowed or divorced. 4.Determine the probability that a randomly selected United States resident 18 years or older is male or widowed.

COMPLEMENTARY EVENTS

RULE OF COMPLEMENTARY EVENTS

VENN DIAGRAM FOR THE COMPLEMENT OF A

EXAMPLE The data in the table below represent the income distribution of households in the US in (Source: US Bureau of the Census) Annual IncomeNumber (in thousands) Annual IncomeNumber (in thousands) Less than $10,00010,023$50,000 to $74,99920,018 $10,000 to $14,9996,995$75,000 to $99,99910,480 $15,000 to $24,99913,994$100,000 to $149,9998,125 $25,000 to $34,99913,491$150,000 to $199,9992,337 $35,000 to $49,99917,032$200,000 or more2,239

EXAMPLE (CONCLUDED) 1.Compute the probability that a randomly selected household earned $200,000 or more in Compute the probability that a randomly selected household earned less than $200,000 in Compute the probability that a randomly selected household earned at least $10,000 in 2000.