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

where P(A and B) denotes the probability that A and B both occur at the same time as an outcome in a trial of a procedure.

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 ONDISJOINT EVENTS

• If two events, A and B, are disjoint, then P(A and B) = 0.

• If events A and B are disjoint, thenP(A or B) = P(A) + P(B).

APPLYING THE ADDITION RULEP(A or B)

Addition Rule

AreA and Bdisjoint

?

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.

EXAMPLEThe data in the chart below represent the marital status of males and females 18 years or older in the US in 1998. 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

25.5 21.0 46.5

Married 58.6 59.3 117.9

Widowed 2.6 11.0 13.6

Divorced 8.3 11.1 19.4

Totals(in millions) 95.0 102.4 197.4

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.

Note that events A and are disjoint. Also, we can be absolutely certain that either A or occurs. So we have

COMPLEMENTARY EVENTS

RULE OF COMPLEMENTARY EVENTS

𝑃 ( 𝐴)+𝑃 ( 𝐴 )=1

VENN DIAGRAM FOR THE COMPLEMENT OF A

EXAMPLEThe data in the table below represent the income distribution of households in the US in 2000. (Source: US Bureau of the Census)Annual Income Number (in

thousands)Annual Income Number (in

thousands)

Less than $10,000 10,023 $50,000 to $74,999 20,018

$10,000 to $14,999 6,995 $75,000 to $99,999 10,480

$15,000 to $24,999 13,994 $100,000 to $149,999 8,125

$25,000 to $34,999 13,491 $150,000 to $199,999 2,337

$35,000 to $49,999 17,032 $200,000 or more 2,239

EXAMPLE (CONCLUDED)

1. Compute the probability that a randomly selected household earned $200,000 or more in 2000.

2. Compute the probability that a randomly selected household earned less than $200,000 in 2000.

3. Compute the probability that a randomly selected household earned at least $10,000 in 2000.