Fault tree probability calculator
AND and OR gates
- AND gate (all events must happen): multiply the probabilities.
- Put in the numbers: 0.1 × 0.2
- Answer: 0.02
- OR gate (at least one event happens): P = P(A) + P(B) − P(A) × P(B)
- Put in the numbers: 0.1 + 0.2 − 0.1 × 0.2
- Work it out: 0.3 − 0.02
- Answer: 0.28
At least once in n tries
- Chance of at least one in n tries: P = 1 − (1 − p)ⁿ
- Put in the numbers: 1 − (1 − 0.01)^100
- Chance of none: 0.99^100 = 0.366
- Answer: 0.634
Formulas and source
- AND: P = P(A) × P(B) × ...
- OR (two events): P = P(A) + P(B) − P(A) × P(B)
- OR (any number): P = 1 − (1 − P₁)(1 − P₂)...
- At least once in n tries: P = 1 − (1 − p)ⁿ
The formula comes from
NUREG-0492, Fault Tree Handbook (U.S. NRC, 1981), Chapter VII(opens nrc.gov in a new tab)“If A and B are independent events, then P(BIA) = P(B) and P(Q) = P(A) + P(B) - P(A) P(B); ... If A and B are independent events, then P(BIA) = P(B), P(AIB) = P(A), and P(Q) = P(A) P(B);”
Formulas and calculators on the BCSP exams
- BCSP gives you no formula sheet, and you can't bring one.
- When a question needs a formula, the formula is written into the question, unless the question is testing whether you know that formula.
- You get an on-screen calculator that works like the TI-30XS, plus a whiteboard. You can't bring your own calculator.
- On the ASP, Mathematical Calculations is its own domain, worth 10% of the exam (ASP11 blueprint).
BCSP on formulas
BCSP, What to Expect When Sitting for a BCSP Exam (item 7)(opens bcsp.org in a new tab)“For any question requiring a mathematical formula, the formula will be included within the exam question (unless the question is specifically testing your knowledge of a formula). You will not be provided or permitted to bring a formula sheet.”
BCSP on calculators
BCSP, What to Expect When Sitting for a BCSP Exam (item 6)(opens bcsp.org in a new tab)“You will be provided an on-screen calculator, emulating the TI-30XS scientific calculator. You will also have access to a white board and dry erase marker. The testing center will not provide a physical calculator or permit candidates to bring their own.”
FAQ
How do you calculate an AND gate?
Multiply the probabilities. All the events have to happen, so the result is smaller than any one of them: 0.1 × 0.2 = 0.02.
How do you calculate an OR gate?
For two events: P = P(A) + P(B) − P(A) × P(B). For 0.1 and 0.2 that is 0.1 + 0.2 − 0.02 = 0.28. For more events, it's easier to find the chance that none happen and subtract from 1: P = 1 − (1 − P₁)(1 − P₂)(1 − P₃)....
Why not just add the probabilities for an OR gate?
Adding counts the case where both happen twice. Subtracting P(A) × P(B) removes the overlap. Adding is only a close estimate when every probability is very small.
What is the chance of at least one event in n tries?
P = 1 − (1 − p)ⁿ. A 1% chance per day over 100 days is 1 − 0.99¹⁰⁰ = 0.634, about 63%, not 100%.
Do these formulas work for any events?
They assume the events are independent, meaning one happening doesn't change the chance of the other. The NRC Fault Tree Handbook states the formulas for independent events.
The OR formula for three or more events and the "at least once" formula follow from applying the two-event rule again and again. This tool is for study.