Advanced Math

Probability Calculator

Clearly defined, finite probability problems: favourable ÷ total outcomes, multiplying probabilities of independent events, adding probabilities of mutually exclusive events, permutations, combinations, and the binomial probability of exactly k successes. Every formula is shown with your values substituted. It does not interpret written word problems.

InputsP(A), nCr, nPr

Problem typeFavourable outcomesTotal equally likely outcomes

Step-by-step

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Fill in the fields and press Solve. The full method will appear here, one numbered step at a time.

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Worked example

A fair coin is tossed 5 times. What is the probability of exactly 2 heads?

1.
Assumptions: 5 independent trials, each a success with the same probability p = 1/2.
2.
Binomial formula: P(X = k) = C(n, k) · p^k · (1 − p)^(n − k).
```
P(X = 2) = C(5, 2) · (1/2)^2 · (1/2)^3
```
3.
C(n, k) counts the ways to place the k successes among the n trials.
```
C(5, 2) = 10
```
4.
Evaluate the powers and multiply.
```
10 × 1/4 × 1/8 = 5/16
```

Answer: P(X = 2) = 5/16

Note: Uses the values you enter; it doesn’t read written questions.

Use “Load worked example” above to run this in the solver.

Related formulas

The formulas this tool uses, each explained with symbols, conditions and its own example in the Formula Library.

Basic probabilityP(A) = favourable outcomes ÷ total outcomes Independent events (AND)P(A and B) = P(A) × P(B)Mutually exclusive events (OR)P(A or B) = P(A) + P(B)PermutationsⁿPᵣ = n! ÷ (n − r)!CombinationsⁿCᵣ = n! ÷ (r!(n − r)!)Binomial probabilityP(X = k) = ⁿCₖ × pᵏ × (1 − p)ⁿ⁻ᵏ