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Probability and Venn diagrams questions

Probability is careful counting, written formally. Venn diagrams keep the counting accurate, the addition rule handles events that overlap, and two special cases, mutually exclusive and independent, account for most of what the exam asks.

6 original questions · 22 marks · the probability and venn diagrams notes · Statistics

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• 2 marks · warm-up

  P(A) = 0.45 and P(B) = 0.35, and A and B are mutually exclusive. Write down P(A ∩ B), and find P(A ∪ B).

  Worked answer

  Mutually exclusive means the events cannot happen together, so P(A ∩ B) = 0. The addition rule then loses its correction term and P(A ∪ B) = 0.45 + 0.35 = 0.8. B1 for P(A ∩ B) = 0, B1 for P(A ∪ B) = 0.8. On a Venn diagram the two circles are drawn apart, with no region between them.

• 3 marks · warm-up

  P(A) = 0.55, P(B) = 0.4 and P(A ∪ B) = 0.75. Find P(A ∩ B), and determine whether A and B are independent.

  Worked answer

  Rearrange the addition rule: P(A ∩ B) = P(A) + P(B) − P(A ∪ B) = 0.55 + 0.4 − 0.75 = 0.2. Now test independence. P(A) × P(B) = 0.55 × 0.4 = 0.22, which does not equal 0.2, so A and B are not independent. M1 for rearranging the addition rule, A1 for 0.2, B1 for the independence test with both values quoted. The conclusion must state the comparison and not only the verdict, so quote both 0.22 and 0.2. The same test in conditional form gives P(A|B) = 0.2/0.4 = 0.5 against P(A) = 0.55, and reaches the same answer.

• 4 marks

  Of 100 students, 55 study art, 40 study music, and 20 study both. Draw the Venn diagram in numbers: art only, music only, both, neither.

  Worked answer

  Put the overlap in first: 20 study both. Art only is then 55 − 20 = 35 and music only is 40 − 20 = 20. Those three regions hold 75, so 25 students sit outside both circles. B1 for the overlap, M1 for subtracting it, A1 for 35 and 20, B1 for the 25 outside. Filling from the middle outwards is the whole technique. The figures 55 and 40 already include the 20, so subtracting the overlap before anything else stops it being counted twice.

• 3 marks

  Using the diagram above, find the probability that a randomly chosen student studies exactly one of the subjects, and the probability that they study neither.

  Worked answer

  Exactly one means art only or music only, so (35 + 20)/100 = 0.55. Neither is the region outside both circles, 25/100 = 0.25. M1 for adding the two single regions, A1 for 0.55, B1 for 0.25. 'Exactly one' deliberately excludes the overlap, which is what makes it a different question from P(A ∪ B) = 0.75. Read the wording before reaching for a region.

• 3 marks · explain

  Explain the difference between mutually exclusive events and independent events.

  Worked answer

  Mutually exclusive events cannot both happen, so their circles do not overlap and P(A ∩ B) = 0. Independent events are those where one happening tells you nothing about the other, so P(A ∩ B) = P(A)P(B). B1 for mutual exclusivity, B1 for independence, B1 for why they cannot hold together. The two ideas are distinct properties rather than opposites. For events of non-zero probability they cannot hold together, because mutual exclusivity is an extreme form of dependence, with knowing that A occurred ruling B out entirely — and a pair of events may easily be neither exclusive nor independent.

• 7 marks · stretch

  Of 60 people surveyed, 30 like tea, 25 like coffee and 22 like juice. 12 like tea and coffee, 10 like coffee and juice, 8 like tea and juice, and 5 like all three. Find the number who like none of the three drinks, and the probability that a person chosen at random likes exactly one drink.

  Worked answer

  Fill the three-circle diagram from the centre outwards. The middle holds 5. Tea and coffee but not juice is 12 − 5 = 7; coffee and juice but not tea is 10 − 5 = 5; tea and juice but not coffee is 8 − 5 = 3. Now the single regions. Tea only is 30 − 7 − 3 − 5 = 15, coffee only is 25 − 7 − 5 − 5 = 8, and juice only is 22 − 3 − 5 − 5 = 9. Those seven regions total 52, so 60 − 52 = 8 people like none of the three. Exactly one drink covers 15 + 8 + 9 = 32 people, a probability of 32/60 = 8/15, or 0.533. The inclusion-exclusion formula gives the same total: 30 + 25 + 22 − 12 − 10 − 8 + 5 = 52. B1 for the centre, M1 A1 for the three pairwise regions, A1 for the three single regions, A1 for the 8 who like none, M1 A1 for the probability. Order is everything here. Subtracting the triple overlap from each pair before touching the single regions is what keeps every person counted exactly once, and working outwards in the other direction always double counts.

The same practice on paper: the printable workbook for this topic, questions and a worked answer book.

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