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.

Confusion about independence has a famous name in psychology: the “gambler's fallacy,” vividly illustrated at a Monte Carlo casino in 1913, when the roulette wheel landed on black 26 times in a row. Gamblers, certain that red was somehow “due,” bet increasingly large sums on red as the streak grew, losing millions of francs in today's money, even though each spin was completely independent of the last and red remained just as likely (or unlikely) as it always had been. The multiplication rule taught on this page is the precise mathematical antidote to that fallacy – it shows exactly why past independent results can never shift the probability of the next one.

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

A coin is tossed and a die is rolled. Find P(heads and a 3).

P(heads) = 1/2. P(3) = 1/6. Events are independent (separate experiments).
P(heads and 3) = 1/2 × 1/6 = 1/12.

A bag contains 3 red and 7 blue balls. A ball is drawn, its colour noted, and replaced. A second ball is then drawn. Find P(red, then red).

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.

A fair coin is tossed three times. Find P(all three heads).

Each toss is independent. P(H) = 1/2 each time.
P(HHH) = 1/2 × 1/2 × 1/2 = 1/8 = 0.125.

P(A) = 0.6 and P(B) = 0.4 where A and B are independent. Find P(A and B) and P(neither A nor B).

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: The Multiplication Rule

Multiplication Rule