Probability • Topic 4 of 5

Independent Events

Two events are independent when one does not affect the other — flipping a coin, then rolling a die, for instance. The probability that both happen is the product of their separate probabilities: P(A and B) = P(A) × P(B). So two coins both landing heads is 1/2 × 1/2 = 1/4, and two dice both showing 6 is 1/6 × 1/6 = 1/36. Extend the rule to more events by multiplying all the individual probabilities. The key check is that the events truly do not influence one another; if they do, the events are dependent and the second probability changes. Multiply, do not add.

A probability tree for two coin tosses giving four outcomes HH, HT, TH and TTIndependent eventsH ½T ½H ½T ½H ½T ½HH ¼HT ¼TH ¼TT ¼Independent: multiply along branches

✅ Solved examples

1. Two coins are tossed. P(both heads)?
1/2 × 1/2 = 1/4.
2. A die is rolled and a coin tossed. P(even and heads)?
1/2 × 1/2 = 1/4.
3. Two dice are rolled. P(both show 6)?
1/6 × 1/6 = 1/36.
4. Three coins are tossed. P(all heads)?
(1/2)³ = 1/8.

✏️ Practice — try these, take hints as needed

1. Two coins are tossed. P(both tails)?
Multiply 1/2 × 1/2.
—
—
1/4.
2. A spinner of 4 sections is spun twice. P(section 1 both times)?
1/4 × 1/4.
—
—
1/16.
3. Two dice are rolled. P(both even)?
P(even) = 1/2 each.
1/2 × 1/2.
—
1/4.
4. A coin and a die. P(heads and a 3)?
1/2 × 1/6.
—
—
1/12.
5. Four coins are tossed. P(all heads)?
(1/2)⁴.
—
—
1/16.

📝 Topic test — 8 questions

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