Formulas used
- P(A) = favorable / total for equally likely outcomes.
- P(Aᶜ) = 1 − P(A) for the complement event.
- P(A∪B) = P(A) + P(B) − P(A∩B) for a union.
- P(A∩B) = P(A)·P(B) (independent) or P(A)·P(B|A) (dependent).
- P(A|B) = P(A∩B) / P(B) when P(B) > 0.
- P(A|B) = P(B|A)·P(A)/P(B) (Bayes’ theorem).
- P(X=k) = C(n,k)·p^k·(1−p)^(n−k) for a binomial model.
- P(X≥1) = 1 − (1−p)^n for at least one success.
How to use this calculator
- Choose a calculation type that matches your scenario.
- Enter probabilities as decimals between 0 and 1.
- For counts, ensure favorable does not exceed total.
- Press Submit to display results above the form.
- Review the steps to confirm the rule and inputs.
- Use CSV or PDF to share or archive the outcome.
FAQs
1) When should I use counts instead of probabilities?
Use counts when outcomes are equally likely and you can list favorable and total outcomes. Use probabilities when you already know rates, forecasts, or model assumptions.
2) What does “independent events” mean here?
Independence means event A happening does not change the chance of B. In that case, the joint probability equals P(A) multiplied by P(B).
3) Why does the union formula subtract the intersection?
Adding P(A) and P(B) counts outcomes where both happen twice. Subtracting P(A∩B) removes the double-counting, giving the correct probability for “A or B”.
4) What if I don’t know P(A∩B) for the union?
If events are disjoint, set P(A∩B)=0. If they are independent, you can estimate P(A∩B)=P(A)·P(B). Otherwise, you need data or assumptions to estimate overlap.
5) How do I interpret conditional probability P(A|B)?
It is the probability of A after you know B happened. It “renormalizes” to the B cases only, so P(B) must be greater than zero.
6) When is Bayes’ theorem useful?
Bayes helps update a prior belief P(A) using new evidence B. It combines a likelihood P(B|A) with the overall evidence rate P(B) to compute the updated probability P(A|B).
7) Do binomial formulas assume identical trials?
Yes. Binomial calculations assume each trial is independent with the same success probability p. If p changes across trials, consider a different model or compute probabilities per trial.