Probability Calculator: Bayes' Theorem, Conditional Probability, Independent Events and Expected Value

Bayes, conditional probability and expected value.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose Bayes, two events, or a distribution.

For Bayes enter how common the condition is and the test's sensitivity and false positive rate. For two events enter P(A), P(B) and P(A and B).

For a distribution enter up to five values with their probabilities, which must add to 1.

Formulas used

  • P(D|+) = \frac{P(+|D)P(D)}{P(+)}
  • P(A\cup B) = P(A) + P(B) - P(A\cap B)
  • P(A|B) = \frac{P(A\cap B)}{P(B)}
  • E(X) = \sum xP(x)

Worked example

With the default numbers in the calculator above:

  1. Chance of a positive test (total probability): 9.81 %
  2. Chance of having the condition given a positive test (Bayes' theorem): 9.1743 %
  3. The same in natural frequencies for 10 000 people: 100 have it, of whom 90 test positive; of the 9900 who do not, 891 also test positive
  4. Chance of not having it given a negative test: 99.889 %

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want? Two events A and B.

  1. Probability of A or B (the addition rule): 0.7
  2. Probability of A given B: 0.5
  3. Probability of B given A: 0.4
  4. Neither A nor B happens: 0.3
  5. Are A and B independent? They are if P(A and B) = P(A) x P(B): 0.2 equals P(A and B): the events are independent
  6. Are they mutually exclusive? They are if they cannot both happen: No: they can both happen

Common questions

What is Bayes' theorem?

It turns the probability of a result given a condition into the probability of the condition given the result.

Why is a positive test often wrong for a rare condition?

Most people do not have the condition, so even a small false positive rate produces many more false positives than true ones.

How do I check if two events are independent?

They are independent if P(A and B) equals P(A) times P(B).

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any first-year probability and statistics text.

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