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**Probability: Venn Diagrams**

Johanna McHardy Mount Albert Grammar School

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**16 girls in a class 7 hockey players 9 softball players 4 play both**

How many play neither sport?

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16 H S 3 4 5 4

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**24 boys live in Manukau Rd 13 soccer players 12 cricket players**

7 play both How many play neither sport?

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24 C S 5 6 7 6

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**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?

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24 C P 7 4 7 6

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**30 boys in a class All play rugby or basketball 19 play rugby**

15 play basketball How many play both games?

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30 B R 11 15 4

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**23 girls in a class 13 can type 12 take woodwork 7 do neither**

How many can type and take woodwork?

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23 W T 3 4 9 7

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**30 men 17 use Pong after-shave 19 use Macho after-shave 4 use neither**

How many use both?

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30 M P 9 7 10 4

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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.

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A class of 40 students … 40 Cats Birds Dogs

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**Everyone liked at least one pet.**

40 Cats Birds Dogs

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**10 students liked cats and birds but not dogs.**

40 Cats Birds 10 Dogs

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**2 students liked birds and dogs but not cats.**

40 Cats Birds 10 2 Dogs

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**12 students liked cats and birds.**

40 Cats Birds 10 2 2 Dogs

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**8 students liked cats and dogs.**

40 Cats Birds 10 2 2 6 Dogs

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**All together, 28 liked cats, 19 liked dogs and 15 liked birds.**

40 Cats Birds 10 10 1 2 2 6 9 Dogs

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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?

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**there were a total of 24 geckoes …**

Grey Tailless Chewed on

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**Only one of the geckoes was grey, chewed on, and tailless**

24 Grey Tailless 1 Chewed on

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**two were grey and tailless but not chewed on;**

24 Grey Tailless 2 1 Chewed on

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**all of the geckoes were at least one of "grey", "tailless", and "chewed on",**

24 Grey Tailless 2 1 2 Chewed on

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

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**twelve geckoes that had dropped their tails: a + b = 9**

24 Grey Tailless 1 2 a 1 b 2 Chewed on

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**fifteen geckoes that he'd chewed: c + b = 12**

24 Grey Tailless 1 2 a 1 b 2 c Chewed on

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**3 unknowns need 3 equations: a + b = 9; c + b = 12; a + b + c = 18**

24 Grey Tailless 1 2 a 1 b 2 c Chewed on

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**tailless and chewed on but not grey = 3**

24 Grey Tailless 1 2 6 1 3 2 9 Chewed on

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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?

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**100 students were interviewed**

PE BIO ENG

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**4 took all three subjects.**

100 PE BIO 4 ENG

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

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

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

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**28 took PE, 31 took BIO, 42 took ENG**

100 PE BIO 5 20 13 4 2 6 20 30 ENG

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**a) How many students took none of the three subjects**

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

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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

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**students surveyed liked at least one of the following three fruits**

62 Apricots Bananas e b a g f d c Cantaloupes

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**34 liked apricots 30 liked bananas 33 liked cantaloupes**

62 Apricots Bananas e b a g f Keep the letters I alphabetic order as it helps to see the patterns. d c Cantaloupes

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**62 e b a g f d c Apricots Bananas Cantaloupes**

11 liked apricots and bananas 15 liked bananas and cantaloupes 17 liked apricots and cantaloupes 62 Apricots Bananas e b a g f See how the g’s line up- it helps d c Cantaloupes

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**62 e b a g f d c Apricots Bananas Cantaloupes**

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

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62 Apricots Bananas 3 b a 8 7 9 c Cantaloupes

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62 Apricots Bananas 3 12 14 8 7 9 9 Cantaloupes

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**a. How many students liked apricots, but not bananas or cantaloupes**

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

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**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? Mixed up messages

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**3 had a hamburger, soft drink and ice-cream.**

90 Hamburger Soft drink 3 Look for all three first Ice-cream

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**5 had a hamburger and a soft drink.**

90 Hamburger Soft drink 2 3 Then look for two at a time Ice-cream

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**10 had a soft drink and ice-cream.**

90 Hamburger Soft drink 2 3 7 Ice-cream

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**8 had a hamburger and ice-cream.**

90 Hamburger Soft drink 2 3 7 5 Ice-cream

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

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

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

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

Ice-cream

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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?

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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.

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

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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.

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Equations

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Equations

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10 students take 3 classes

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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?

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

relish

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**38 had all three condiments.**

20 256 ketchup mustard a e b 38 f d c relish

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

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**68 had ketchup and mustard**

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

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**62 had mustard but no relish**

20 256 ketchup mustard a 30 32 38 f 46 c relish

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

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**(a) The number sold with relish only is**

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

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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?

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

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

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

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**How many dogs are black with long hair but do not have short tails? 3**

24 Black Short tails 6 2 1 1 2 3 9 Long Hair

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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?

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

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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? Basic Problem

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

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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 and Chorus, 20 are in both the Band and 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? Easy

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120 Band Chorus 21 29 25 8 11 12 14 Drama

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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

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

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**English, but not Business studies or Computer Science 15**

English or Business studies but not Computer Science 45 English or Business studies 315

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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?

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79 Maths Geography 8 6 29 2 4 14 16 History

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**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

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