Conditional probability
Conditional Probability
Conditional Probability is a topic in Probability & Statistics in the California Common Core State Standards. It is aligned to Standard S-CP.6, which requires students to find and interpret conditional probability as the probability of A given B.
Conditional probability \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\) is the chance of \(A\) given that \(B\) has occurred.
Theory
Conditional probability and its rearrangement:
How to find a conditional probability
- Identify the condition \(B\) (what is given).
- Restrict attention to outcomes in \(B\).
- Divide the joint probability by \(P(B)\).
- Or count directly: favorable-and-\(B\) over total-in-\(B\).
Divide the joint probability by \(P(B)\).
| \(P(A\mid B)\) | \(=\) | \(\dfrac{P(A\cap B)}{P(B)}\) |
| \(=\) | \(\dfrac{0.2}{0.5}=0.4\) |
Restrict to the \(12\) face cards; \(4\) are kings.
| \(P(\text{king}\mid\text{face})\) | \(=\) | \(\dfrac{4}{12}=\dfrac13\) |
Condition on the \(30\) who play a sport.
| \(P\) | \(=\) | \(\dfrac{18}{30}=\dfrac35\) |
Rearrange the conditional formula.
| \(P(A\cap B)\) | \(=\) | \(P(B)\,P(A\mid B)\) |
| \(=\) | \(0.6\times0.5=0.3\) |
Common pitfalls
Frequently asked questions
What is conditional probability?
The probability of one event given that another has occurred, \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\).
How does conditioning change the sample space?
It restricts it to outcomes where the condition holds, so you count only within \(B\).
Is \(P(A\mid B)\) the same as \(P(B\mid A)\)?
No, they are generally different.
How is conditional probability related to the multiplication rule?
Rearranging gives \(P(A\cap B)=P(B)P(A\mid B)\), the general multiplication rule.