Algebra 2
Statistics and probability
Conditional probability and independence
20 practice questions
0 video lessons
Theory + worked examples
Theory
Conditional probability is the chance of \(A\) given that \(B\) occurred:
\[P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}.\]
Events are independent when \(P(A\mid B)=P(A)\) — knowing \(B\) doesn't change \(A\).
Given \(B\) shrinks the sample space to \(B\).
Conditioning on \(B\) restricts to \(B\).
Conditional probability and independence.
Conditional and independence:
\[P(A\mid B)=\dfrac{P(A\cap B)}{P(B)},\qquad \text{independent: }P(A\mid B)=P(A)\]
Independent events multiply directly: \(P(A\cap B)=P(A)P(B)\).
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.
Example 1 — From counts
\(P(A\cap B)=0.2,\ P(B)=0.5\). Find \(P(A\mid B)\).
Solution
Divide by \(P(B)\).
| \(P(A\mid B)\) | \(=\) | \(\dfrac{0.2}{0.5}=0.4\) |
Example 2 — Cards
A card is a face card. Find \(P(\text{king}\mid\text{face})\).
Solution
Restrict to the \(12\) face cards.
| \(\dfrac{4}{12}\) | \(=\) | \(\dfrac13\) |
Example 3 — Test independence
\(P(A)=0.4,\ P(A\mid B)=0.4\). Are \(A,B\) independent?
Solution
Independence means \(P(A\mid B)=P(A)\).
| \(0.4\) | \(=\) | \(0.4\ \checkmark\) |
Yes — they are independent.
Example 4 — Everyday example
Give an example of dependent events.
Solution
Drawing two cards without replacement — the first draw changes the second's probability.
Common pitfalls
\(P(A\mid B)\neq P(B\mid A)\) in general.
Divide by \(P(B)\), the condition.
Independent means the conditional equals the original, not zero overlap.
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.
← Previous subtopic
Probability rules (addition, multiplication)
Next subtopic →
Two-way tables for probability
More in Statistics and probability