Conditional Probability and Bayes' Theorem
P(A|B) = P(A and B)/P(B) for P(B) non-zero, and Bayes' theorem inverts the conditioning using the total probability rule.
What is Conditional Probability and Bayes' Theorem?
P(A|B) = P(A and B)/P(B) for P(B) non-zero, and Bayes' theorem inverts the conditioning using the total probability rule.
Key formula / rule: Independence means P(A and B) = P(A) P(B).
Key points
- Compute conditional probabilities from a described situation.
- Apply Bayes' theorem to invert conditional probabilities.
Common exam trap
Confusing P(A|B) with P(B|A).
Definitions
- Term
Conditional Probability and Bayes' Theorem
- Meaning
P(A|B) = P(A and B)/P(B) for P(B) non-zero, and Bayes' theorem inverts the conditioning using the total probability rule.
- Term
Conditional Probability and Bayes' Theorem — explanation
- Meaning
Conditioning restricts the sample space to B, which is why the denominator changes. Bayes' theorem is the standard route from likelihoods to posterior probabilities in partition problems.
Learning objectives
Compute conditional probabilities from a described situation.
Apply Bayes' theorem to invert conditional probabilities.
Formulae
- Key point
Independence means P(A and B) = P(A) P(B).
- Key point
Total probability sums P(A|Bi) P(Bi) over a partition.
- Key point
Bayes' theorem reverses the direction of conditioning.
Prerequisites
MAT-U5-PS-T1-S1-C1
Common mistakes
Confusing P(A|B) with P(B|A).
Treating mutually exclusive events as independent.
Keywords
Conditional
Probability
Bayes'
Theorem
Practice preview
Given P(A) = 0.4, P(B) = 0.5, and P(A and B) = 0.2, find P(A|B).…
easy
For two independent events A and B, if P(A) = 0.7 and P(B) = 0.4, what is P(A|B)?…
easy
If P(A) = 0.6, P(B) = 0.3, and P(A and B) = 0.18, what is P(B|A)?…
easy