Independent events and the Multiplication Rule
Independent Events and the Multiplication Rule
Independent Events and the Multiplication Rule is a topic in Probability & Statistics in the Common Core State Standards. It is aligned to Standard S-CP.2, which requires students to understand independence and apply the multiplication rule for independent events.
Independent events do not affect each other, so the probability that both occur is the product \(P(A)\,P(B)\).
Theory
Two events are independent if one occurring does not change the probability of the other. For independent events:
The multiplication rule (independent events):
How to use the multiplication rule
- Confirm the events are independent.
- Find each event's probability.
- Multiply them for \(P(A\cap B)\).
- To test independence, compare \(P(A\cap B)\) with \(P(A)P(B)\).
The flips are independent, so multiply.
| \(P(HH)\) | \(=\) | \(\dfrac12\cdot\dfrac12=\dfrac14\) |
Independent events multiply.
| \(P\) | \(=\) | \(\dfrac16\cdot\dfrac12=\dfrac1{12}\) |
Check whether \(P(A\cap B)=P(A)P(B)\).
| \(P(A)P(B)\) | \(=\) | \(0.4\times0.5=0.2\) |
| \(P(A\cap B)\) | \(=\) | \(0.2\) |
They are equal, so the events are independent.
Multiply the independent probabilities.
| \(P\) | \(=\) | \(\left(\dfrac13\right)^3=\dfrac1{27}\) |
Common pitfalls
Frequently asked questions
What are independent events?
Events where one occurring does not change the probability of the other.
What is the multiplication rule?
For independent events, \(P(A\cap B)=P(A)P(B)\).
How do you test for independence?
Check whether \(P(A\cap B)=P(A)P(B)\); if so, the events are independent.
Are independent and mutually exclusive the same?
No. Mutually exclusive events cannot both occur, so they are actually dependent.