Independent Events - When One Does Not Affect the Other
Two events are independent if the occurrence of one event has absolutely no effect on the probability of the other. Tossing a coin twice is a classic example – whether the first toss lands heads or tails tells us nothing about what the second toss will produce.
The Multiplication Rule for Independent Events
If A and B are independent events:
P(A and B) = P(A) × P(B)
This extends to any number of independent events:
P(A and B and C) = P(A) × P(B) × P(C).
How to Recognise Independent Events
- The result of one experiment does not change the conditions of the next.
- Replacing a drawn item before the next draw keeps events independent.
- Two separate, unconnected experiments are always independent.
Worked Examples
P(heads) = 1/2. P(3) = 1/6. Events are independent (separate experiments).
P(heads and 3) = 1/2 × 1/6 = 1/12.
Because the ball is replaced, the second draw is independent of the first.
P(red) = 3/10 each time.
P(red and red) = 3/10 × 3/10 = 9/100 = 0.09.
Each toss is independent. P(H) = 1/2 each time.
P(HHH) = 1/2 × 1/2 × 1/2 = 1/8 = 0.125.
P(A and B) = 0.6 × 0.4 = 0.24.
P(not A) = 1 − 0.6 = 0.4. P(not B) = 1 − 0.4 = 0.6.
P(neither) = 0.4 × 0.6 = 0.24.
Independent vs. Dependent Events
| Independent | Dependent | |
|---|---|---|
| Definition | One event does not affect the other | One event changes the probability of the other |
| Example | Drawing with replacement | Drawing without replacement |
| Rule | P(A and B) = P(A) × P(B) | P(A and B) = P(A) × P(B|A) |
Key Takeaways
- Independent events: the outcome of one does not affect the probability of the other.
- P(A and B) = P(A) × P(B) for independent events.
- Replacement keeps draws independent; no replacement makes them dependent.
- The rule extends to any number of independent events by continued multiplication.
Practice Questions
- A die is rolled twice. Find P(4 on first roll and 4 on second roll).
- A bag has 5 red and 5 blue balls. A ball is drawn, replaced, and a second is drawn. Find P(both blue).
- P(A) = 0.7 and P(B) = 0.3. A and B are independent. Find P(A and B).
- A coin is tossed four times. Find P(all tails).
- Two machines each have a 0.95 probability of working on any given day, independently. Find the probability that both work on the same day.