🎲 Probability & Combinatorics Toolkit

Solve foundational probability questions, evaluate permutations & combinations, and run quick simulations without leaving your browser. Built for classrooms, analytics teams, and exam prep. Every calculator is mobile-first and aligned with CalcArena’s design principles.

1. Basic & Conditional Probability

Covers single-event probability, complements, addition rules, and independence checks.

Single Event Probability

Number of favourable outcomes *

Total number of equally-likely outcomes *

Probability P(A): —; —
Complement P(A'): —; Probability event does not occur.

Two Events (A & B)

Enter probabilities as percentages between 0 and 100. Intersection P(A ∩ B) ≤ min(P(A), P(B)).

P(A) %

P(B) %

P(A ∩ B) %

Optional: P(B|A) %

Union P(A ∪ B): —
Conditional P(A | B): —
Conditional P(B | A): —
Independence Check: —

2. Bayes’ Theorem & Diagnostic Tests

Compute Positive Predictive Value (PPV) and Negative Predictive Value (NPV) from medical test metrics.

Prevalence (Prior) % *

Sensitivity (True Positive Rate) % *

Specificity (True Negative Rate) % *

Key Outputs

Positive Predictive Value (PPV): —
Negative Predictive Value (NPV): —
False Positive Rate (1 − Specificity): —
False Negative Rate (1 − Sensitivity): —

3. Permutations & Arrangements

Includes nPr, permutations with repeated elements, and circular arrangements.

Distinct Permutations (nPr)

Total elements (n)

Pick (r)

Permutations

—

Permutations with Repetition

Enter counts of identical items separated by commas (e.g. for “BANANA” → 3,2,1).

Total characters (optional)

Unique permutations

—

Circular Permutations

Number of distinct items

(n − 1)! arrangements

—

For restrictions (e.g. no adjacent seats), combine with inclusion–exclusion principles below.

4. Combinations & Binomial Coefficients

Supports large n choose r using log-gamma, plus stars-and-bars and lottery odds.

Standard Combination

Total items (n)

Select (r)

Combinations

—

Stars & Bars (Identical Items)

Identical items

Groups / bins

Ways to distribute

—

Lottery Odds (n choose r)

Compare total combinations with probability of a single ticket winning.

Numbers in pool

Numbers drawn

Total combinations

—

5. Counting Principles & Inclusion–Exclusion

Multiply independent stages, or combine overlapping sets using inclusion–exclusion.

Fundamental Counting Principle

Enter stage counts separated by commas (e.g. license plate with 26 letters and 10 digits → 26,26,26,10,10,10).

Total outcomes

—

Inclusion–Exclusion (Two Sets)

|A|

|B|

|A ∩ B|

Union size

—

Inclusion–Exclusion (Three Sets)

|A|

|B|

|C|

|A ∩ B|

|A ∩ C|

|B ∩ C|

|A ∩ B ∩ C|

Union size

—

6. Probability Experiments & Simulators

Run quick Monte Carlo experiments to compare empirical vs theoretical probabilities.

Coin Toss Simulator

Number of tosses

Results

Heads: —
Tails: —
Difference from 50%: —

Dice Roll Distribution

Number of dice

Rolls per simulation

Frequency by outcome

—

Birthday Paradox Calculator

People in room

Days per year

Probability of shared birthday

—

Monty Hall Simulator

Number of trials

Winning Probability

Stay strategy: —
Switch strategy: —

Switching should approach 66.7% with enough trials.

Formula References

Ross, S. M. (2014). A First Course in Probability, Pearson.
Casella, G., & Berger, R. L. (2002). Statistical Inference, Duxbury.
Tucker, A. (2002). Applied Combinatorics, Wiley.
Rosen, K. H. (2012). Discrete Mathematics and Its Applications, McGraw-Hill.
MIT OpenCourseWare (2018). Probabilistic Systems Analysis and Applied Probability simulations.