Library / Foundations: Maths / FMA-06
Foundations: Maths
Independent events: both and either
Chance that two independent events both happen, or at least one does.
- Ref
- FMA-06
- Rev
- v1
- Field
- Foundations: Maths
- Result
- P(A∩B) [%]
For independent events multiply their probabilities to find the probability that both occur. The complement rule is P(A′) = 1 − P(A).
If one event does not affect the other, the chance of both happening is the product of their separate chances.
Calculator
- At least one happens65 %
- Neither happens35 %
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.
- Both events together is always rarer than either alone.
- For two independent parts working in series, system reliability is the product of the part reliabilities.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (%) | Local step | Change in result |
|---|---|---|---|---|---|---|
| P(A) Probability of A | 5 to 90 % | 2.5 | 45 | 42.5 | +10% | +1.5 pp |
| P(B) Probability of B | 5 to 90 % | 1.5 | 27 | 25.5 | +10% | +1.5 pp |
Sweep of Probability of A
| Probability of A (%) | Probability of both (%) |
|---|---|
| 5 | 2.5 |
| 15.63 | 7.813 |
| 26.25 | 13.13 |
| 36.88 | 18.44 |
| 47.5 | 23.75 |
| 58.13 | 29.06 |
| 68.75 | 34.38 |
| 79.38 | 39.69 |
| 90 | 45 |
Sweep one variable
Variables and how each one contributes
Probability of both
What the equation gives you. Everything else feeds it.
Probability of A
Chance that A happens. Directly proportional.
Probability of B
Chance that B happens. Directly proportional.
Worked example with the values above
- Convert to fractions: 0.3 and 0.5.
- Multiply: 0.3 × 0.5 = 0.15.
- As a percentage 15 %.
Assumptions and limits
- The two events do not influence each other.
- Probabilities are for the same trial or period.
Common mistakes
- Adding probabilities when both must happen.
- Multiplying for mutually exclusive events.
Statistics and uncertainty
Real events are often not independent, and the product rule then misstates the chance. Uncertainty in each probability combines as a relative error: δP/P ≈ √[(δP_A/P_A)² + (δP_B/P_B)²].
Where it is used
- System reliability from component reliabilities.
- Chance of multiple independent failures.
Maths you need
- Probability scale: 0 to 1 or 0 to 100%
- Multiplication of fractions: product of decimals