Library / Foundations: Maths / FMA-30
Foundations: Maths
Conditional probability
Chance of A given that B has already happened.
- Ref
- FMA-30
- Rev
- v1
- Field
- Foundations: Maths
- Result
- P(A|B) [%]
Requires P(B) > 0. For independent events P(A ∩ B) = P(A)P(B). For any events, subtract the overlap when adding probabilities for their union.
Once you know B has happened, the chance of A is the share of B’s outcomes that also include A.
Calculator
%
%
%
- As a fraction0.4
- Probability B does not happen70 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
Conditional probability (%) current value 40 %
P(A∩B)Probability of bothKey driver
6.67100
P(B)Probability of B
2097.3
- Conditioning shrinks the sample space to B alone.
- If P(A|B) equals P(A), then A and B are independent.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (%) | Local step | Change in result |
|---|---|---|---|---|---|---|
| P(A∩B) Probability of both | 2 to 30 % | 6.667 | 100 | 93.33 | +10% | +4 pp |
| P(B) Probability of B | 5 to 60 % | 20 | 97.3 | 77.3 | +10% | −3.64 pp |
Sweep of Probability of both
| Probability of both (%) | Conditional probability (%) |
|---|---|
| 2 | 6.667 |
| 5.5 | 18.33 |
| 9 | 30 |
| 12.5 | 41.67 |
| 16 | 53.33 |
| 19.5 | 65 |
| 23 | 76.67 |
| 26.5 | 88.33 |
| 30 | 100 |
Sweep one variable
Conditional probability (%)Hover, touch, or use the arrow keys on the chart
Variables and how each one contributes
P(A|B)
Conditional probability
What the equation gives you. Everything else feeds it.
P(A∩B)
Probability of both
Chance that A and B both happen. Directly proportional.
P(B)
Probability of B
Chance that B happens. Inversely proportional.
Worked example with the values above
- P(A ∩ B) = 12 %, P(B) = 30 %.
- Divide: 12 / 30 = 0.4.
- P(A | B) = 40 %.
Assumptions and limits
- P(B) is greater than zero.
- Probabilities refer to the same population.
Common mistakes
- Dividing by P(A) instead of P(B).
- Confusing P(A|B) with P(B|A).
Statistics and uncertainty
Conditional probabilities estimated from small samples are noisy; the error grows when P(B) is rare since few cases qualify.
Where it is used
- Detection probabilities given a cue (false alarms and misses).
- Fault diagnosis given a symptom.
Maths you need
- Probability: 0 to 1 scale
- Venn diagrams: overlap of events