Direct answer
Conditional probability restricts attention to outcomes inside the conditioning event: P(A|B)=P(A∩B)/P(B) when P(B)>0.
Visual explanation
Conditioning changes the denominator
What this calculation tells you
Conditional probability measures the share of outcomes satisfying A among outcomes already known to satisfy B. It is a change of reference set, not an extra event added after the calculation.
Translate the question into a numerator and denominator in words before using symbols. The numerator must satisfy both conditions; the denominator must contain every outcome allowed after the given information is applied.
Where it is used
Quality
Find a defect share within one product or process subgroup.
Research
Describe outcomes within an explicitly defined stratum.
Reliability
Update component-event probabilities after observed system states.
Education
Connect tables, Venn regions, and probability trees to one denominator rule.
When this guide helps
- Reading a two-way frequency table.
- Separating joint and conditional probability.
- Explaining why reversed conditionals differ.
- Checking whether a subgroup denominator is complete.
Start with the statistical question
Translate the question into a numerator and denominator in words before using symbols. The numerator must satisfy both conditions; the denominator must contain every outcome allowed after the given information is applied.
A full population rectangle narrows to the B region, then highlights the part also in A. The denominator visibly changes from the whole population to B.
Worked example
If 30 of 100 records are in B and 18 of those 30 are also in A, then P(A|B)=18/30=0.60. Dividing 18 by 100 instead answers the joint-probability question P(A∩B).
Assumptions that carry the result
Events, counts, and populations must use the same inclusion rules and observation unit. Time order alone does not prove dependence, and observational conditioning does not automatically create a causal comparison.
Interpret the result without overreaching
A conditional association may reflect selection, common causes, or measurement differences. The arithmetic updates a reference set; it does not establish why the probabilities differ.
- Reversing P(A|B) and P(B|A).
- Using the original population as the denominator after conditioning.
- Treating conditional association as causation.
Worked case: cards given a face card
From a 52-card deck, condition on the card being J, Q or K.
There are 12 face cards and 4 kings, so P(king|face) = 4/12 = 1/3.
The denominator is the 12-card conditioned sample space.
Using 4/52 would ignore the supplied information.
Reproduce this worked caseOpen Conditional Probability Calculator
Worked case: reverse the condition
Find P(face card|king).
Every king is a face card, so 4/4 = 1.
The reversed conditional probability is 1, not 1/3.
P(A|B) and P(B|A) are generally different.
Reproduce this worked caseOpen Conditional Probability Calculator
conditional probability: compare assumptions, not just answers
Write the condition after the bar and rebuild the denominator before calculating. A zero-probability condition is undefined in elementary finite probability.
| Case | Calculation focus | Interpretation |
|---|---|---|
| King | face | 4/12 | 1/3 |
| Face | king | 4/4 | 1 |
conditional probability: calculation checklist
- Condition event identified
- Intersection counted
- Condition denominator nonzero
- Direction not reversed
- Sampling model stated
Practical questions
Frequently asked questions
Is P(A|B) the same as P(B|A)?
Usually not. They use different conditioning sets and therefore different denominators.
What if P(B)=0?
The elementary ratio is undefined because there are no probability-mass outcomes inside B to use as a denominator.
Does conditioning prove B caused A?
No. It describes probability within B; causal interpretation requires design and assumptions beyond the ratio.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.