Venn diagrams for probability Problem set A Worked solutions

Of 50 families, 30 have a dog and 20 have a cat. 8 families have both. One family is chosen at random. Find the probability that the family has

1. (a) a dog but not a cat.

   • Both: 8. First set only: 30 − 8 = 22
   • \(P=\frac{22}{50}=\frac{11}{25}\)

2. (b) a cat but not a dog.

   • Second set only: 20 − 8 = 12
   • \(P=\frac{12}{50}=\frac{6}{25}\)

In a class of 36 students, 22 play football and 15 play hockey. 7 play both. Find the probability that a student chosen at random plays neither sport.

• Only the first: 22 − 7 = 15. Only the second: 15 − 7 = 8
• Neither: 36 − 15 − 7 − 8 = 6
• \(P(\text{neither})=\frac{6}{36}=\frac{1}{6}\)

The Venn diagram shows the 60 members of a gym. C is the set who attend classes and W is the set who use the weights room. The numbers are numbers of members. A member is chosen at random. Find the probability that the member

1. (a) attends classes.

   • Inside the first circle: 17 + 11 = 28
   • \(P=\frac{28}{60}=\frac{7}{15}\)

2. (b) attends classes but does not use the weights room.

   • Only the first circle: 17
   • \(P=\frac{17}{60}\)

3. (c) is in neither set.

   • Outside both circles: 9
   • \(P=\frac{9}{60}=\frac{3}{20}\)

In a school of 90 students, 14 study both French and Spanish, 20 study neither language and 33 study French only.

1. (a) How many students study Spanish only?

   • 90 − 14 − 20 − 33 = 23

2. (b) A student is chosen at random. Find the probability that the student studies Spanish.

   • All of Spanish: 23 + 14 = 37
   • \(P=\frac{37}{90}\)

A survey asks 120 people what they watch. 70 watch drama, 55 watch comedy and 25 watch both.

1. (a) Draw a Venn diagram.

   • Both: 25. Only drama: 70 − 25 = 45. Only comedy: 55 − 25 = 30
   • Neither: 120 − 45 − 25 − 30 = 20

2. (b) Find the probability that a person chosen at random does neither.

   • \(P=\frac{20}{120}=\frac{1}{6}\)

3. (c) Find the probability that a person chosen at random does exactly one.

   • Exactly one: 45 + 30 = 75
   • \(P=\frac{75}{120}=\frac{5}{8}\)

Of 200 shoppers, 36 buy both bread and milk, 24 buy neither and 75 buy milk only. A shopper is chosen at random. Find the probability that the shopper buys bread. Give your answer as a decimal.

• Bread only: 200 − 36 − 24 − 75 = 65
• All who buy bread: 65 + 36 = 101
• \(P=\frac{101}{200}\) = 0.505

In a village, the probability that a house has broadband is 0.6 and the probability that it has a landline is 0.5. The probability that it has neither is 0.1.

1. (a) Find the probability of a house having both.

   • P(at least one) = 1 − 0.1 = 0.9
   • 0.6 + 0.5 − P(both) = 0.9
   • P(both) = 0.2

2. (b) Find the probability of a house having broadband but no landline.

   • First thing only = 0.6 − 0.2 = 0.4

In a club of 80 members, 20 do both kayaking and climbing and 12 do neither. The number who do kayaking only is 8 more than the number who do climbing only. Find the probability that a member chosen at random does climbing. Give your answer as a fraction.

• Let the number who do climbing only be x. Then kayaking only is x + 8.
• x + (x + 8) + 20 + 12 = 80, so 2x = 40 and x = 20
• All who do climbing: 20 + 20 = 40
• \(P=\frac{40}{80}=\frac{1}{2}\)