Chapter 7Chapter 7Sets & ProbabilitySets & Probability

Section 7.2Section 7.2

Applications of Venn DiagramsApplications of Venn Diagrams

Shade the region corresponding to A B .

A B

Shade the region corresponding to A B .

A B

Step 1: Shade the region corresponding to A .

Shade the region corresponding to A B .

A B

Step 2: Shade the region corresponding to B .

The intersection of A B is the overlapping region that has been shaded twice.

Shade the region corresponding to A B .

A B

A B

Use a Venn diagram to shade in the Use a Venn diagram to shade in the region corresponding to the indicated set.region corresponding to the indicated set.

a.) B a.) B A A

b.) B b.) B (A (A C C ))

c.) A c.) A (B (B C) C)

A survey of 200 people yielded the following A survey of 200 people yielded the following information: 94 owned a VCR, 127 owned a information: 94 owned a VCR, 127 owned a microwave oven, and 78 owned both. How microwave oven, and 78 owned both. How many people owned the following?many people owned the following?

a.) a VCR or a microwave ovena.) a VCR or a microwave oven

b.) a VCR but not a microwave b.) a VCR but not a microwave

c.) a microwave oven but not a VCRc.) a microwave oven but not a VCR

d.) neither a VCR nor a microwave ovend.) neither a VCR nor a microwave oven

A survey of 200 people yielded the following information: 94 owned a A survey of 200 people yielded the following information: 94 owned a VCR, 127 owned a microwave oven, and 78 owned both. VCR, 127 owned a microwave oven, and 78 owned both.

n(U) = 200n(U) = 200

n(V) = 94n(V) = 94

n(M) = 127n(M) = 127

n(V n(V M) = 78 M) = 78

V M

78 4916

57

A survey of 200 people yielded the following information: 94 A survey of 200 people yielded the following information: 94 owned a VCR, 127 owned a microwave oven, and 78 owned owned a VCR, 127 owned a microwave oven, and 78 owned both. How many people owned…both. How many people owned…

V M

78 4916

57

a.) a VCR or microwave oven?

n (V M) = 16 + 78 + 49 = 143

b.) a VCR, but not a microwave oven?

n (V M) = 16

c.) a microwave oven, but not a VCR?

n (M V) = 49

d.) neither a VCR nor a microwave oven?

n (M V) = 57

The records of 1492 high school graduates were examined, The records of 1492 high school graduates were examined, and the following information was obtained: 1072 took biology, and the following information was obtained: 1072 took biology, and 679 took geometry. If 271 of those who took geometry did and 679 took geometry. If 271 of those who took geometry did not take biology, how many graduates took the following?not take biology, how many graduates took the following?

a.) both classesa.) both classes

b.) at least one of the classesb.) at least one of the classes

c.) biology but not geometryc.) biology but not geometry

d.) neither classd.) neither class

In a survey, 674 adults were asked what television In a survey, 674 adults were asked what television programs they had recently watched. The following programs they had recently watched. The following information was obtained: 226 watched neither the Big information was obtained: 226 watched neither the Big Game nor the New Movie, and 289 watched the New Game nor the New Movie, and 289 watched the New Movie. If 183 of those who watched the New Movie did not Movie. If 183 of those who watched the New Movie did not watch the Big Game, how many of the surveyed adults watch the Big Game, how many of the surveyed adults watched the following?watched the following?

a.) both programsa.) both programs

b.) at least one programb.) at least one program

c.) the Big Gamec.) the Big Game

d.) the Big Game but not the New Movied.) the Big Game but not the New Movie

Recently, U2, Sting, and Lollapalooza each toured the Recently, U2, Sting, and Lollapalooza each toured the United States. A large group of college students was United States. A large group of college students was surveyed, and the following information was obtained: 1533 surveyed, and the following information was obtained: 1533 saw U2, 1127 saw Sting, 581 saw Lollapalooza, 219 saw all saw U2, 1127 saw Sting, 581 saw Lollapalooza, 219 saw all three, 1853 saw none, 766 saw only U2, 654 saw U2 and three, 1853 saw none, 766 saw only U2, 654 saw U2 and Sting, and 92 saw Sting and Lollapalooza but not U2.Sting, and 92 saw Sting and Lollapalooza but not U2.

a.) What percent of the college students saw all three a.) What percent of the college students saw all three acts?acts?

b.) What percent of the college students saw at least two b.) What percent of the college students saw at least two of the acts?of the acts?

A kindergarten teacher wants to do an A kindergarten teacher wants to do an activity about common household pets. She activity about common household pets. She begins the activity by asking her class the begins the activity by asking her class the following questions and records the following questions and records the children’s answers.children’s answers.

““How many of you have a dog?”How many of you have a dog?” 15 15

““How many of you have a cat?”How many of you have a cat?” 1010

““How many of you have both?”How many of you have both?” 77

How many of the children had a dog or a cat?How many of the children had a dog or a cat?

Union Rule for Sets Union Rule for Sets (General Rule)(General Rule)

n(A n(A B) = n(A) + n(B) – n(A B) = n(A) + n(B) – n(A B) B)

n(D) = 15n(D) = 15 n(C) = 10n(C) = 10 n(D n(D C) = 7 C) = 7

n(D n(D C) = n(D) + n(C) – n(D C) = n(D) + n(C) – n(D C) C)

n(D n(D C) = 15 + 10 – 7 C) = 15 + 10 – 7

n(D n(D C) = 18 C) = 18

Union Rule for Disjoint SetsUnion Rule for Disjoint Sets

n(A n(A B) = n(A) + n(B) B) = n(A) + n(B)

Note: Use this rule only when the two sets do Note: Use this rule only when the two sets do

not have any elements in common.not have any elements in common.

To be on the safe side, you can To be on the safe side, you can alwaysalways use the General Union Rule for Sets.use the General Union Rule for Sets.

A company that specializes in language tutoring lists the A company that specializes in language tutoring lists the following information concerning its English-speaking following information concerning its English-speaking employees: 23 speak German, 25 speak French, 31 speak employees: 23 speak German, 25 speak French, 31 speak Spanish, 43 speak Spanish or French, 38 speak French or Spanish, 43 speak Spanish or French, 38 speak French or German, 46 speak German or Spanish, 8 speak Spanish, German, 46 speak German or Spanish, 8 speak Spanish, French, and German, and 7 office workers and secretaries French, and German, and 7 office workers and secretaries speak English only.speak English only.

a.) What percent of the employees speak at least one a.) What percent of the employees speak at least one language other than English?language other than English?

b.) What percent of the employees speak at least two b.) What percent of the employees speak at least two languages other than English?languages other than English?