5.2 Venn Diagrams

5.2 Venn Diagrams

Key Facts

• Probabilities can be represented on Venn diagrams.
• Let A be an event, B be an event, and S represent the sample space.
• Intersection: The shaded area representing event A and B, denoted as A ∩ B.
• Union: The shaded area representing event A or B, denoted as A ∪ B.
• Complement: The shaded area representing the event not A.
  • The probability of not A happening is: P(not \, A) = 1 - P(A)
• Let P represent probability. Then:
  • 0 ≤ P ≤ 1
  • The sum of all probabilities equals 1.

Exam-Style Question 1

• Scenario: A group of 275 people at a music festival were asked if they play guitar, piano, or drums.
  • 1 person plays all three instruments.
  • 65 play guitar and piano.
  • 10 play piano and drums.
  • 30 play guitar and drums.
  • 15 play piano only.
  • 20 play guitar only.
  • 35 play drums only.

Part A: Draw a Venn diagram to represent this information.

• Let G = event play guitar, P = event play piano, and D = event play drums.
• Start with the intersection of all three events (G ∩ P ∩ D), which is 1.
• Guitar and Piano (G ∩ P): 65 total, so 65 - 1 = 64.
• Piano and Drums (P ∩ D): 10 total, so 10 - 1 = 9.
• Guitar and Drums (G ∩ D): 30 total, so 30 - 1 = 29.
• Piano only: 15.
• Guitar only: 20.
• Drums only: 35.
• Outside: 275 - (20 + 15 + 35 + 64 + 1 + 9 + 29) = 102.

Part B

• A festival-goer is chosen at random.

Part 1: Find the probability that the person chosen plays piano.

• P(P) = (64 + 1 + 9 + 15) / 275 = 89/275

Part 2: Plays at least two of guitar, piano, and drums.

• P(at \, least \, two) = (29 + 64 + 9 + 1) / 275 = 103/275

Part 3: Plays exactly one of the instruments.

• P(exactly \, one) = (20 + 15 + 35) / 275 = 70/275

Part 4: Plays none of the instruments.

• P(none) = 102/275

Exam-Style Question 2

• The Venn diagram shows probabilities of students studying mathematics (M), physics (P), and history (H).
• Given: P(M) = P(P)
• Find the values of P and Q.

Solution

• P(M) = 0.32 + P
• P(P) = P + Q + 0.07
• Since P(M) = P(P), then 0.32 + P = P + Q + 0.07
• Rearrange to find Q: Q = 0.32 - 0.07 = 0.25
• Probabilities add up to 1:
  • 0.32 + P + Q + 0.07 + 0.13 + 0.1 = 1
  • P + Q + 0.62 = 1
• Substitute Q = 0.25:
  • P + 0.25 + 0.62 = 1
  • P + 0.87 = 1
• Solve for P: P = 1 - 0.87 = 0.13