Theoretical Probability with Two Independent Events

Multiply the individual probabilities directly — no table required.

▶ Practice this concept

🧱 See It

Before, finding a two-event probability meant building a whole outcomes table and counting favorable pairings. There's a shortcut: for independent events (where one doesn't affect the other), just multiply the two individual probabilities directly.

✏️ Draw It

Spinner A has 4 equal sections; spinner B has 2 equal sections:

Red
Blue
Green
Yellow
Win
Lose

P(Blue) = 1/4. P(Win) = 1/2. Spinning both together, neither one affects the other.

🔢 Write It

P(A and B) = P(A) × P(B)  —  for independent events only

For Blue and Win: P(Blue and Win) = 1/4 × 1/2 = 1/8

This shortcut only works when the events are truly independent. Always ask first: "does the first event change what's possible for the second?"

💡 Worked Examples

Example 1 — Flip two coins. Find P(both Heads).

  1. P(Heads on coin 1) = 1/2
  2. P(Heads on coin 2) = 1/2
  3. Multiply: 1/2 × 1/2 = 1/4
  4. Answer: 1/4

Example 2 — Spin a 5-section spinner (1 section is "Star") and roll a die. Find P(Star and rolling a 6).

  1. P(Star) = 1/5
  2. P(rolling a 6) = 1/6
  3. Multiply: 1/5 × 1/6 = 1/30
  4. Answer: 1/30

🧠 Quick Reference

P(A and B) = P(A) × P(B) — but only for INDEPENDENT events. Multiply straight across like any fraction multiplication, no common denominator needed. If one event changes the odds for the other (like drawing marbles without putting them back), this shortcut does NOT apply.
👪 For Parents & Tutors

BC Curriculum (Level 3 / Grade 8, Data & Probability): "Calculate theoretical probability with two independent events."

Watch for: applying the multiply-straight-across shortcut without checking independence first — if the events aren't truly independent (e.g. drawing two marbles from a bag without replacing the first), the individual probabilities change between events and this rule breaks. Have him explicitly say out loud "does the first event change what's possible for the second?" before multiplying.

Comes up again: Level 4 (Grade 9) statistics work on data reliability and bias builds on being comfortable distinguishing independent from dependent events.