Conditional probability and independence
Conditional Probability and Independence
Conditional Probability and Independence is a topic in Statistics & Probability in the Common Core State Standards. It is aligned to Standard S-CP.3, which requires students to understand conditional probability and independence and interpret them in context.
Conditional probability \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\) is the chance of \(A\) given \(B\); events are independent when \(P(A\mid B)=P(A)\).
Theory
Events are independent when \(P(A\mid B)=P(A)\) — knowing \(B\) doesn't change \(A\).
Conditional and independence:
How to work with conditionals
- Identify the given event \(B\).
- Divide the joint probability by \(P(B)\).
- To test independence, compare \(P(A\mid B)\) with \(P(A)\).
- If equal, the events are independent.
Divide by \(P(B)\).
| \(P(A\mid B)\) | \(=\) | \(\dfrac{0.2}{0.5}=0.4\) |
Restrict to the \(12\) face cards.
| \(\dfrac{4}{12}\) | \(=\) | \(\dfrac13\) |
Independence means \(P(A\mid B)=P(A)\).
| \(0.4\) | \(=\) | \(0.4\ \checkmark\) |
Yes — they are independent.
Drawing two cards without replacement — the first draw changes the second's probability.
Common pitfalls
Frequently asked questions
What is conditional probability?
The probability of one event given that another has occurred.
What is the formula?
\(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\).
When are two events independent?
When \(P(A\mid B)=P(A)\).
Are dependent events common?
Yes — e.g. drawing without replacement.