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Learn Index · Probability and Combinatorics Grade 9–10

Probability of compound events

The chance of two or more events happening together or one of them happening.

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Combining events

A compound event combines two or more events, such as "roll a 6 and flip heads" or "draw a king or a heart".

"And": both happen

For independent events (one doesn’t affect the other), use the multiplication rule:

P(A and B) = P(A) × P(B)

Example: Roll a die and flip a coin. P(6 and heads)?

1/6 × 1/2 = 1/12

Check with the sample space: 6 × 2 = 12 outcomes, and only one is (6, H) ✓.

Example: P(three heads in a row) = (1/2)³ = 1/8

"Or": at least one happens

Use the addition rule:

P(A or B) = P(A) + P(B) − P(A and B)

The subtraction stops outcomes in both A and B from being counted twice.

Example: Draw one card. P(king or heart)?

  • P(king) = 4/52
  • P(heart) = 13/52
  • P(king of hearts) = 1/52
  • P = (4 + 13 − 1)/52 = 16/52 = 4/13

Mutually exclusive events

Mutually exclusive events can’t both happen. Then P(A and B) = 0, so:

P(A or B) = P(A) + P(B)

Example: Roll a die. P(2 or 5) = 1/6 + 1/6 = 1/3.

"At least one": use the complement

P(at least one) = 1 − P(none)

Example: Flip 3 coins. P(at least one head)?

  • P(no heads) = (1/2)³ = 1/8
  • P(at least one) = 1 − 1/8 = 7/8

Tables and tree diagrams

Organized tables and tree diagrams show the sample space for compound events.

Two dice: P(sum = 8) = 5/36 (the pairs 2–6, 3–5, 4–4, 5–3, 6–2).

Worked examples

Example 1: P(rolling an even number and spinning red on a spinner that is ¼ red)? ½ × ¼ = 1/8.

Example 2: P(drawing a red card or a face card)? 26/52 + 12/52 − 6/52 = 32/52 = 8/13.

Example 3: A basketball player makes 80% of free throws. P(making two in a row), assuming independence? 0.8 × 0.8 = 0.64.

Common mistakes

  • Adding when you should multiply ("and" means multiply for independent events).
  • Forgetting to subtract the overlap in "or" problems.
  • Assuming events are independent when they aren’t (like drawing cards without replacement).

Try it yourself

  1. P(heads and rolling a 3)?
  2. P(rolling a 1 or a 6)?
  3. P(drawing an ace or a spade)?
  4. P(at least one 6 in two rolls)?
  5. A weather app says 30% chance of rain each day, independently. P(no rain on both days)?

Answers: 1) 1/12 2) 1/3 3) 16/52 = 4/13 4) 1 − 25/36 = 11/36 5) 0.49

Quick recap

  • "And" with independent events: multiply.
  • "Or": add and subtract the overlap; for mutually exclusive events, just add.
  • "At least one" = 1 − P(none).