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Independent and mutually exclusive events

Multiplying probabilities for independent events and adding for mutually exclusive ones.

  • Define and explain Independent and mutually exclusive events in your own words
  • Use key terms such as Mutually exclusive accurately
  • Apply what you have learned to new examples and questions
  • Avoid the common mistakes learners make with this topic

This lesson focuses on Independent and mutually exclusive events: multiplying probabilities for independent events and adding for mutually exclusive ones.

Definition: Independent and mutually exclusive events

Multiplying probabilities for independent events and adding for mutually exclusive ones.

Key ideas

The addition rule

For mutually exclusive events, P(A or B) = P(A) + P(B). Rolling a 1 or a 6 on a fair die gives 1/6 + 1/6 = 2/6 = 1/3. This only works when the events cannot both happen.

The multiplication rule

For independent events, P(A and B) = P(A) × P(B). Two heads on two fair coin tosses gives 1/2 × 1/2 = 1/4. For dependent events such as drawing without replacement, use the new totals after each draw.

Key term — Mutually exclusive: Two events that cannot happen at the same time, such as rolling a 2 and rolling a 5 on one throw; their probabilities add.

Probability without replacement

A bag holds 3 red and 2 blue counters. Two are drawn without replacement. Find the probability both are red.

First draw: P(red) = 3/5. Second draw: one red is gone, so 2 red remain out of 4 counters: P(red given red) = 2/4 = 1/2. Multiply: 3/5 × 1/2 = 3/10.

Answer: 3/10 (0.3 or 30%).

Common mistakes
  • Adding probabilities of events that overlap The addition rule needs mutually exclusive events; drawing a red card or a king double-counts the king of hearts.
  • Multiplying when events are dependent Without replacement the second probability changes — use the reduced totals, or draw a tree diagram.

Practice

A fair six-sided die is rolled. Find P(rolling an even number).
Count the even faces: 2, 4, 6.

P = 3/6 = 1/2.

A fair coin is tossed twice. Find P(two heads).
Tosses are independent: multiply.

P = 1/2 × 1/2 = 1/4.

P(it rains tomorrow) = 0.3. Find P(it does not rain).
Use the complement: probabilities sum to 1.

P = 1 − 0.3 = 0.7.

A spinner has sectors 1, 2, 3, 4 (equally likely). Find P(spinning a 1 or a 2).
The outcomes are mutually exclusive: add.

P = 1/4 + 1/4 = 1/2.

Quick check

Independent and mutually exclusive events — quick check

Which of these best defines "Mutually exclusive"?

Two events that cannot happen at the same time, such as rolling a 2 and rolling a 5 on one throw; their probabilities add.

Two dice are rolled. Find P(total of 7).

6 favourable pairs out of 36: P = 6/36 = 1/6.

A bag holds 4 green and 6 yellow sweets. One sweet is taken at random. Find P(green).

P = 4/10 = 2/5.
Key takeaways
  • Independent and mutually exclusive events: multiplying probabilities for independent events and adding for mutually exclusive ones.
  • The addition rule: For mutually exclusive events, P(A or B) = P(A) + P(B).
  • Independent: Two events are independent when one does not affect the other, such as two coin tosses; multiply their probabilities.
  • Watch out for: adding probabilities of events that overlap
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