Probability Venn Diagrams Johanna Mc Hardy Mount Albert

Probability: Venn Diagrams Johanna Mc. Hardy Mount Albert Grammar School

16 girls in a class • • 7 hockey players 9 softball players 4 play both How many play neither sport?

16 H S 3 4 5 4

24 boys live in Manukau Rd • • 13 soccer players 12 cricket players 7 play both How many play neither sport?

24 C S 6 7 5 6

24 people attend a party • 4 eat pizza but not chicken • 6 eat neither chicken nor pizza • 7 eat both • How many people eat chicken but not pizza?

24 C P 4 7 7 6

30 boys in a class • All play rugby or basketball • 19 play rugby • 15 play basketball • How many play both games?

30 B R 15 4 11 0

23 girls in a class • • 13 can type 12 take woodwork 7 do neither How many can type and take woodwork?

23 W T 4 9 3 7

30 men • 17 use Pong aftershave • 19 use Macho aftershave • 4 use neither • How many use both?

30 M P 7 9 10 4

Question One A class of 40 students completed a survey on pets they like. The choices were dogs, cats and birds. • Everyone liked at least one pet. • 10 students liked cats and birds but not dogs. • 2 students liked birds and dogs but not cats. • 12 students liked cats and birds. • 8 students liked cats and dogs. • All together, 28 liked cats, 19 liked dogs and 15 liked birds. Represent this information on a Venn diagram.

A class of 40 students … Cats Birds Dogs 40

Everyone liked at least one pet. Cats 0 Birds Dogs 40

10 students liked cats and birds but not dogs. Cats Birds 10 0 Dogs 40

2 students liked birds and dogs but not cats. Cats Birds 10 2 0 Dogs 40

12 students liked cats and birds. Cats Birds 10 2 2 0 Dogs 40

8 students liked cats and dogs. Cats Birds 10 6 0 2 2 Dogs 40

All together, 28 liked cats, 19 liked dogs and 15 liked birds. Cats Birds 10 10 6 0 2 1 2 9 Dogs 40

Question 2 Suppose I discovered that my cat had a taste for the adorable little geckoes that live in the bushes and vines in my yard, back when I lived in Arizona. In one month, suppose he deposited the following on my carpet: six grey geckoes, twelve geckoes that had dropped their tails in an effort to escape capture, and fifteen geckoes that he'd chewed on a little. Only one of the geckoes was grey, chewed on, and tailless; two were grey and tailless but not chewed on; two were grey and chewed on but not tailless. If there were a total of 24 geckoes left on my carpet that month, and all of the geckoes were at least one of "grey", "tailless", and "chewed on", how many were tailless and chewed on but not grey?

there were a total of 24 geckoes … Grey Tailless Chewed on 24

Only one of the geckoes was grey, chewed on, and tailless Grey Tailless 1 Chewed on 24

two were grey and tailless but not chewed on; Grey Tailless 2 1 Chewed on 24

all of the geckoes were at least one of "grey", "tailless", and "chewed on", Grey Tailless 2 1 2 0 Chewed on 24

six grey geckoes Grey Tailless 1 2 1 2 0 Chewed on 24

twelve geckoes that had dropped their tails: a + b = 9 Grey Tailless 1 2 1 2 0 a b Chewed on 24

fifteen geckoes that he'd chewed: c + b = 12 Grey Tailless 1 2 1 2 0 c a b Chewed on 24

3 unknowns need 3 equations: a + b = 9; c + b = 12; a + b + c = 18 Grey Tailless 1 2 1 2 0 c a b Chewed on 24

tailless and chewed on but not grey = 3 Grey Tailless 1 2 1 2 0 9 6 3 Chewed on 24

Question 3 100 students were interviewed 28 took PE, 31 took BIO, 42 took ENG, 9 took PE and BIO, 10 took PE and ENG, 6 took BIO and ENG, 4 took all three subjects. a) How many students took none of the three subjects? b) How many students took PE but not BIO or ENG? c) How many students took BIO and PE but not ENG?

100 students were interviewed PE BIO ENG 100

4 took all three subjects. PE BIO 4 ENG 100

9 took PE and BIO. PE 5 BIO 4 ENG 100

10 took PE and ENG PE 5 BIO 4 6 ENG 100

6 took BIO and ENG PE BIO 5 4 6 2 ENG 100

28 took PE, 31 took BIO, 42 took ENG PE 5 13 4 6 20 30 20 BIO 2 ENG 100

a) How many students took none of the three subjects? 20 b) How many students took PE but not BIO or ENG? 13 c) How many students took BIO and PE but not ENG? 5

Question 4 A group of 62 students were surveyed, and it was found that each of the students surveyed liked at least one of the following three fruits: apricots, bananas, and cantaloupes. 34 liked apricots. 30 liked bananas. 33 liked cantaloupes. 11 liked apricots and bananas. 15 liked bananas and cantaloupes. 17 liked apricots and cantaloupes. 19 liked exactly two of the following fruits: apricots, bananas, and cantaloupes

students surveyed liked at least one of the following three fruits Apricots e a b g d 0 c Bananas f Cantaloupes 62

34 liked apricots 30 liked bananas 33 liked cantaloupes Apricots e a b g d 0 c Bananas f Cantaloupes 62

11 liked apricots and bananas 15 liked bananas and cantaloupes 17 liked apricots and cantaloupes Apricots e a b g d 0 c Bananas f Cantaloupes 62

19 liked exactly two of the following fruits: apricots, bananas, and cantaloupes Apricots e a b g d 0 c Bananas f Cantaloupes 62

Apricots 3 a b 8 9 0 c Bananas 7 Cantaloupes 62

Apricots 3 14 8 9 0 9 12 Bananas 7 Cantaloupes 62

a. How many students liked apricots, but not bananas or cantaloupes? 14 b. How many students liked cantaloupes, but not bananas or apricots? 9 c. How many students liked all of the following three fruits: apricots, bananas, and cantaloupes? 8 d. How many students liked apricots and cantaloupes, but not bananas? 9

Question 5 90 students went to a school carnival. • 3 had a hamburger, soft drink and ice-cream. • 24 had hamburgers. • 5 had a hamburger and a soft drink. • 33 had soft drinks. • 10 had a soft drink and ice-cream. • 38 had ice-cream. • 8 had a hamburger and ice-cream. • How many had nothing?

3 had a hamburger, soft drink and ice-cream. 90 Hamburger Soft drink 3 Ice-cream

5 had a hamburger and a soft drink. 90 Hamburger Soft drink 2 3 Ice-cream

10 had a soft drink and ice cream. 90 Hamburger Soft drink 2 3 7 Ice-cream

8 had a hamburger and ice cream. 90 Hamburger Soft drink 2 3 5 7 Ice-cream

24 had hamburgers. 90 Hamburger Soft drink 14 2 3 5 7 Ice-cream

33 had soft drinks. 90 Hamburger Soft drink 14 2 3 5 21 7 Ice-cream

38 had ice-cream. 90 Hamburger Soft drink 14 2 3 5 23 21 7 Ice-cream

How many had nothing? 15 90 Hamburger Soft drink 14 2 3 5 15 23 21 7 Ice-cream

Question 6 200 students are surveyed. 80 take Math, 60 take History, 140 take English, 40 take Math and History, 30 take History and English, and 20 take Math and English. How many students take all 3 classes? What assumption do you have to make in order to answer this question?

Question 6 200 students are surveyed. 80 take Math, 60 take History, 140 take English, 40 take Math and History, 30 take History and English, and 20 take Math and English. How many students take all 3 classes? What assumption do you have to make in order to answer this question? Assumption is that all students take at least one of the three subjects.

200 students are surveyed Maths History a e g 0 d c b f English 200

Equations 200 students are surveyed. 80 take Math, 60 take History, 140 take English, 40 take Math and History, 30 take History and English, and 20 take Math and English.

Equations

Equations

10 students take 3 classes

Question 7 Superburger sells hamburgers with the choice of ketchup, mustard and relish. One day they sold 256 hamburgers; 140 had mustard, 140 had ketchup, 84 had ketchup and relish, 62 had mustard but no relish, 68 had ketchup and mustard, 38 had all three condiments and 20 had none. • (a) The number sold with relish only is? • (b) The number sold with no relish is?

256 burgers sold; 20 had none. ketchup mustard a 20 e d g c b f relish 256

38 had all three condiments. ketchup mustard a 20 e d 38 c b f relish 256

84 had ketchup and relish ketchup mustard a 20 e 46 38 c b f relish 256

68 had ketchup and mustard ketchup mustard a 20 30 b 46 38 c f relish 256

62 had mustard but no relish ketchup mustard a 20 30 32 46 38 c f relish 256

140 had mustard, ketchup mustard 26 20 30 32 46 38 24 40 relish 256

(a) The number sold with relish only is? 24 (b) The number sold with no relish is? 88 ketchup mustard 26 20 30 32 46 38 24 40 relish 256

Question 8 Twenty-four dogs are in a kennel. Twelve of the dogs are black, six of the dogs have short tails, and fifteen of the dogs have long hair. There is only one dog that is black with a short tail and long hair. Two of the dogs are black with short tails and do not have long hair. Two of the dogs have short tails and long hair but are not black. If all of the dogs in the kennel have at least one of the mentioned characteristics, how many dogs are black with long hair but do not have short tails?

Easy start Black a 0 2 b 1 c 1 Short tails 2 Long Hair 24

Equations: Black a 0 2 b 1 c 1 Short tails 2 Long Hair 24

Equations: Black 6 0 2 3 1 9 1 Short tails 2 Long Hair 24

How many dogs are black with long hair but do not have short tails? 3 Black 6 0 2 3 1 9 1 Short tails 2 Long Hair 24

Question 9 There are 150 students at Seward High School. 66 students play baseball, 45 play basketball, and 42 play soccer. 27 students play exactly two sports, and three students play all three of the sports. How many of the 150 students play none of the three sports?

150 Baseball Basketball 66 -a-b-3 n a b 45 -b-c-3 3 c 42 -a-c-3 Soccer

Question 10 Hypothetically, there are 178 grade 12 students at FLC High School, 92 have enrolled in Data Management, 71 have enrolled in Advanced Functions, and 40 have enrolled in Calculus. The math students include 14 who are taking both Data Management and Advanced Functions, 18 are taking Data Management and Calculus, 11 are taking Advanced Functions and Calculus. Lastly there are 8 brave souls taking all three maths. How many grade 12 students at FLC High school not enrolled in any math class?

178 Data Management 68 10 6 10 8 19 54 Advanced functions 3 Calculus

Question 11 A certain school has three performing arts extracurricular activities: Band, Chorus, or Drama. Students must participate in at least one, and may participate in two or even in all three. There are 120 students in the school. There are 70 students in Band, 73 in the Chorus, and 45 in the Drama. Furthermore, 37 students are in both the Band Chorus, 20 are in both the Band the Drama, and 8 students are in all three groups. Twenty-five students are just in the chorus, not in anything else. How many students participate in only the drama?

Band Chorus 21 0 29 12 8 14 25 11 Drama 120

Question 12 At Dawnview High there are 400 Grade 11 learners. 270 do Computer Science, 300 do English and 50 do Business studies. All those doing Computer Science do English, 20 take Computer Science and Business studies and 35 take English and Business studies. Using a Venn diagram, calculate the probability that a pupil drawn at random will take: • English, but not Business studies or Computer Science • English but not Business studies • English or Business studies but not Computer Science • English or Business studies

Computer Science 0 85 English 250 0 20 15 15 Business Studies 400

• English, but not Business studies or Computer Science 15 • English but not Business studies 265 • English or Business studies but not Computer Science 45 • English or Business studies 315

Question 13 There are 79 Grade 10 learners at school. All of these take some combination of Maths, Geography and History. The number who take Geography is 41; those who take History is 36; and 30 take Maths. The number who take Maths and History is 16; the number who take Geography and History is 6, and there are 8 who take Maths only and 16 who take History only. • Draw a Venn diagram to illustrate all this information. • How many learners take Maths and Geography but not History? • How many learners take Geography only? • How many learners take all three subjects?

79 Maths Geography 8 0 6 14 2 16 29 4 History

• How many learners take Maths and Geography but not History? 6 • How many learners take Geography only? 29 • How many learners take all three subjects? 2