Mutually Exclusive Events

\( P(A\cap B)=0,\quad P(A\cup B)=P(A)+P(B) \)
Probability GCSE
Question 8 of 20
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A spinner has 5 equal sectors. Find P(1 or 2).

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
Add 1/5 + 1/5.

Explanation

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Statement

Two events are mutually exclusive if they cannot happen at the same time. In probability, this means the events have no overlap. For example, when rolling a dice, the outcomes “rolling a 2” and “rolling a 5” are mutually exclusive — only one can occur in a single roll. The rules are:

\[ P(A \cap B) = 0, \quad P(A \cup B) = P(A) + P(B) \]

Here, \(A \cap B\) represents “both A and B”, while \(A \cup B\) represents “A or B”.

Why it’s true

  • If two events cannot happen together, the probability of them both occurring at once must be 0.
  • The probability of either occurring is just the sum of their separate probabilities, since there is no overlap to subtract.

Recipe (how to use it)

  1. Identify whether the events are mutually exclusive.
  2. Write down \(P(A)\) and \(P(B)\).
  3. For “A and B”: answer is 0.
  4. For “A or B”: add \(P(A) + P(B)\).

Spotting it

Typical signs of mutually exclusive events are phrases like “cannot happen at the same time”, “either-or outcomes”, or cases where events are clearly separate (such as rolling a dice or drawing a single card).

Common pairings

  • Coin toss: “heads” and “tails”.
  • Dice roll: two different numbers.
  • Card draw: drawing a heart or drawing a spade in one draw.

Mini examples

  1. Given: \(P(A)=0.3\), \(P(B)=0.5\), and A, B are mutually exclusive. Find: \(P(A \cup B)\). Answer: \(0.3 + 0.5 = 0.8\).
  2. Given: When rolling a dice, \(P(2)=1/6\), \(P(5)=1/6\). Find: probability of rolling a 2 or 5. Answer: \(1/6 + 1/6 = 2/6 = 1/3\).

Pitfalls

  • Adding probabilities when events are not mutually exclusive (e.g., drawing a heart and a red card — they overlap).
  • Forgetting to check whether events can happen simultaneously.
  • Misusing the multiplication rule for independent events instead of addition rule here.

Exam strategy

  • Look for keywords: “mutually exclusive”, “either A or B, not both”.
  • If asked for “A and B” in mutually exclusive cases, the answer is always 0.
  • Keep probabilities between 0 and 1 — never above 1 when adding.

Summary

Mutually exclusive events are events that cannot occur at the same time. Their probability rules are simple: the intersection is always 0, and the union is the sum. Recognising mutually exclusive events helps avoid double-counting in probability problems.

Worked examples

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  1. A dice is rolled. Find the probability of getting a 1 or a 4.
    1. \( P(1) = 1/6 \)
    2. \( P(4) = 1/6 \)
    3. Events are mutually exclusive.
    4. \( P(1 or 4) = 1/6 + 1/6 = 2/6 = 1/3 \)
    Answer: 1/3
  2. A card is drawn. Find the probability of getting a heart or a club.
    1. \( P(Heart) = 13/52 = 1/4 \)
    2. \( P(Club) = 13/52 = 1/4 \)
    3. \( P(Heart or Club) = 1/4 + 1/4 = 1/2 \)
    Answer: 1/2
  3. A coin is tossed. Find the probability of getting heads or tails.
    1. \( P(Head) = 1/2 \)
    2. \( P(Tail) = 1/2 \)
    3. \( P(Head or Tail) = 1/2 + 1/2 = 1 \)
    Answer: 1
  4. A dice is rolled. Find the probability of getting a 3 or a 6.
    1. \( P(3) = 1/6 \)
    2. \( P(6) = 1/6 \)
    3. \( P(3 or 6) = 1/6 + 1/6 = 2/6 = 1/3 \)
    Answer: 1/3
  5. From a bag with 5 red and 5 blue counters, find P(Red or Blue).
    1. \( P(Red) = 5/10 = 1/2 \)
    2. \( P(Blue) = 5/10 = 1/2 \)
    3. \( P(Red or Blue) = 1/2 + 1/2 = 1 \)
    Answer: 1
  6. \( Events A and B are mutually exclusive. P(A)=0.4, P(B)=0.3. Find P(A or B). \)
    1. \( P(A or B) = P(A)+P(B) = 0.4 + 0.3 \)
    2. \( Answer = 0.7 \)
    Answer: 0.7
  7. Find P(getting an Ace or a King in one card draw).
    1. \( P(Ace) = 4/52 = 1/13 \)
    2. \( P(King) = 4/52 = 1/13 \)
    3. \( P(Ace or King) = 1/13 + 1/13 = 2/13 \)
    Answer: 2/13
  8. \( Events A and B are mutually exclusive. P(A)=0.25, P(B)=0.5. Find P(A ∩ B). \)
    1. \( For mutually exclusive, P(A ∩ B) = 0. \)
    Answer: 0
  9. A spinner has equal sectors numbered 1 to 4. Find P(landing on 1 or 3).
    1. \( P(1)=1/4 \)
    2. \( P(3)=1/4 \)
    3. \( P(1 or 3)=1/4+1/4=1/2 \)
    Answer: 1/2
  10. \( Events A and B are mutually exclusive. P(A)=0.45, P(B)=0.35. Find P(A or B). \)
    1. \( P(A or B) = P(A)+P(B) = 0.45+0.35=0.8 \)
    Answer: 0.8