Venn diagrams for probability Problem set B One question at a time

Of 60 staff at a college, 34 drive to work and 28 cycle to work. 12 do both, on different days. One member of staff is chosen at random. Find the probability that the person

1. (a) drives but does not cycle.

   • Both: 12. First set only: 34 − 12 = 22
   • \(P=\frac{22}{60}=\frac{11}{30}\)

2. (b) cycles but does not drive.

   • Second set only: 28 − 12 = 16
   • \(P=\frac{16}{60}=\frac{4}{15}\)

At a youth club of 45 members, 27 like chess and 20 like draughts. 9 like both games. Find the probability that a member chosen at random likes neither game.

• Only the first: 27 − 9 = 18. Only the second: 20 − 9 = 11
• Neither: 45 − 18 − 9 − 11 = 7
• \(P(\text{neither})=\frac{7}{45}\)

The Venn diagram shows the 60 children at a holiday camp. R is the set who go riding and H is the set who go hiking. The numbers are numbers of children. A child is chosen at random. Find the probability that the child

1. (a) goes riding.

   • Inside the first circle: 19 + 8 = 27
   • \(P=\frac{27}{60}=\frac{9}{20}\)

2. (b) goes riding but does not go hiking.

   • Only the first circle: 19
   • \(P=\frac{19}{60}\)

3. (c) is in neither set.

   • Outside both circles: 12
   • \(P=\frac{12}{60}=\frac{1}{5}\)

In a sixth form of 110 students, 18 study both Physics and Chemistry, 25 study neither subject and 40 study Physics only.

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

   • 110 − 18 − 25 − 40 = 27

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

   • All of Chemistry: 27 + 18 = 45
   • \(P=\frac{45}{110}=\frac{9}{22}\)

A poll asks 150 voters which newspapers they read. 84 read a daily paper, 65 read a weekly paper and 31 read both.

1. (a) Draw a Venn diagram.

   • Both: 31. Only daily: 84 − 31 = 53. Only weekly: 65 − 31 = 34
   • Neither: 150 − 53 − 31 − 34 = 32

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

   • \(P=\frac{32}{150}=\frac{16}{75}\)

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

   • Exactly one: 53 + 34 = 87
   • \(P=\frac{87}{150}=\frac{29}{50}\)

Of 240 customers at a stall, 42 buy both tea and coffee, 30 buy neither and 95 buy coffee only. A customer is chosen at random. Find the probability that the customer buys tea. Give your answer as a decimal.

• Tea only: 240 − 42 − 30 − 95 = 73
• All who buy tea: 73 + 42 = 115
• \(P=\frac{115}{240}=\frac{23}{48}\) = 0.479

At a sports centre, the probability that a member swims is 0.7 and the probability that a member plays squash is 0.45. The probability that a member does neither is 0.05.

1. (a) Find the probability of a member doing both.

   • P(at least one) = 1 − 0.05 = 0.95
   • 0.7 + 0.45 − P(both) = 0.95
   • P(both) = 0.2

2. (b) Find the probability of a member swimming but not playing squash.

   • First thing only = 0.7 − 0.2 = 0.5

In a group of 90 players, 16 play both tennis and badminton and 14 play neither. The number who play tennis only is 6 more than the number who play badminton only. Find the probability that a player chosen at random plays badminton. Give your answer as a fraction.

• Let the number who do badminton only be x. Then tennis only is x + 6.
• x + (x + 6) + 16 + 14 = 90, so 2x = 54 and x = 27
• All who do badminton: 27 + 16 = 43
• \(P=\frac{43}{90}\)