Independent Events and the Multiplication Rule
Simple Explanation
Two events are independent if the outcome of one has no effect on the probability of the other. For independent events, the probability that both happen is the product of their individual probabilities: P(A∩B) = P(A) × P(B).
Why Do We Need It?
Many real situations involve repeated or unrelated events (like separate coin flips or dice rolls) — recognizing independence lets you find combined probabilities with simple multiplication.
Formula
The Multiplication Rule for Independent Events
P(A ∩ B) = P(A) × P(B) (A, B independent)
When two events do not affect each other's outcome, the probability that both happen equals the product of their individual probabilities.
- P(A ∩ B)
- — the probability that both A and B happen
- independent
- — the outcome of A has no effect on the probability of B, and vice versa
When to use it: Whenever two events are independent and you need the probability that both occur.
Worked Example
Find the probability of two independent events
A coin is tossed and a die is rolled. Find the probability of getting heads AND rolling a 6.
Why Does This Work?
If A and B are independent, then B occurring within the reduced sample space "given A" is exactly as likely as B occurring overall — so the fraction of A's outcomes that also satisfy B is just P(B), which is why multiplying the two whole-sample-space probabilities gives the correct joint probability.
Real-Life Example
Independent quality checks
A factory has two independent inspection machines, each catching a defect with a known probability.
The probability that both machines catch the same defect (if they operate independently) is the product of their individual detection probabilities — an application of the multiplication rule to system reliability.
Practice
Two fair coins are tossed. Find the probability that both show heads.
MediumCommon mistake
Applying the multiplication rule (P(A)×P(B)) to events that are NOT actually independent — first always confirm whether one event affects the other before multiplying their raw probabilities.
Quick Review
- Independent events: the outcome of one has no effect on the other.
- P(A∩B) = P(A) × P(B), for independent events only.
- Always confirm independence before applying this rule.