5.3: Mutually Exclusive and Independent Events

• Mutually Exclusive Events

  • Definition: Events that have no outcomes in common, meaning they cannot occur at the same time.

• For mutually exclusive events A and B:

  • P(A \text{ and } B) = 0

  • This is because there is no overlap between the events.

  • P(A \text{ or } B) = P(A) + P(B)

Independent Events

• Definition: When one event has no effect on the outcome of another event.

• Test for Independent Events:

  • For independent events A and B:

    • P(A \text{ and } B) = P(A) \cdot P(B)

General Probability Rules

• Probability values range from 0 to 1:

  • 0 \leq P \leq 1

• The total probability of all possible outcomes is 1.

Exam-Style Question 1

• Scenario: A Venn diagram shows the number of children in a playgroup who like playing with bricks (B), action figures (F), or trains (T).

Part A: Identify Mutually Exclusive Events

• From the Venn diagram, the probability of B and T is zero. This means that no children like playing with both bricks and trains simultaneously.

  • P(B \text{ and } T) = 0

• Conclusion: Events B and T are mutually exclusive.

Part B: Determine Independence of Events

• Check if events B and F (bricks and action figures) are independent.

• Calculations:

  • P(B) = \frac{4}{21} (3 + 1 = 4 children like bricks out of a total of 21 children)

  • P(F) = \frac{11}{21} (1 + 4 + 6 = 11 children like action figures out of a total of 21 children)

  • P(B) \cdot P(F) = \frac{4}{21} \cdot \frac{11}{21} = \frac{44}{441}

  • P(B \text{ and } F) = \frac{1}{21} (1 child likes both bricks and action figures)

• Comparison:

  • \frac{44}{441} \neq \frac{1}{21}

  • Since P(B \text{ and } F) \neq P(B) \cdot P(F), the events B and F are not independent.

• Conclusion: The events B and F are not independent.

Exam-Style Question 2

• Scenario: A Venn diagram shows the probabilities of members of a social club taking part in charitable activities: Archery (A), Raffle (R), and Fun Run (F).

• Given: The probability that a member takes part in the archery competition or the raffle is 0.6.

Part A: Find the Values of x and y

• Decode the given information mathematically:

  • P(A \text{ or } R) = 0.6

• Since A and R are mutually exclusive (no overlap):

  • P(A \text{ or } R) = P(A) + P(R)

• From the Venn diagram:

  • P(A) = 0.2

  • P(R) = 0.25 + x

• Equation:

  • 0.2 + 0.25 + x = 0.6

  • 0.45 + x = 0.6

  • x = 0.6 - 0.45 = 0.15

• Find y:

  • The total probability of all events must equal 1.

  • 0.2 + 0.25 + x + y + 0.1 = 1

  • x + y + 0.55 = 1

  • Substitute the value of x:

    • 0.15 + y + 0.55 = 1

    • y + 0.7 = 1

    • y = 1 - 0.7 = 0.3

• Results:

  • x = 0.15

  • y = 0.3

Part B: Show R and F are Not Independent

• Calculations:

  • P(R) = 0.25 + x = 0.25 + 0.15 = 0.4

  • P(F) = x + y = 0.15 + 0.3 = 0.45

  • P(R) \cdot P(F) = 0.4 \cdot 0.45 = 0.18

  • P(R \text{ and } F) = x = 0.15

• Comparison:

  • 0.18 \neq 0.15

  • Since P(R \text{ and } F) \neq P(R) \cdot P(F), the events R and F are not independent.

• Conclusion: The events R and F are not independent.