Probability Calculator
with Step-by-Step Explanation
Describe any probability problem — single event, conditional, binomial, normal distribution — and get a full solution explained in plain language, powered by MathGPT Free.
No account required · All probability types · Full AI explanations
Describe your problem in plain text — or use exact values like P(A) = 0.3 for structured problems
Step 2 — Apply the Addition Rule:
P(A or B) = P(A) + P(B) − P(A and B) = 0.6 + 0.4 − 0.2 = 0.8
Step 3 — Conditional Probability:
P(A|B) = P(A and B) / P(B) = 0.2 / 0.4 = 0.5
Final Answers: P(A or B) = 0.8 · P(A|B) = 0.5
Every probability problem type, in one place
Single Event Probability
Calculate the likelihood of one outcome from a defined sample space. Covers classical, empirical, and theoretical probability calculations.
Conditional Probability
Find P(A|B) when one event depends on another. Applies Bayes’ theorem, multiplication rule, and the conditional formula automatically.
Binomial Distribution
Compute exact, cumulative (≤ or ≥), and complementary binomial probabilities. Shows the full nCk formula with each substitution.
Normal Distribution
Convert raw scores to z-scores, find area under the normal curve, and interpret results. Covers one-tailed and two-tailed probability.
Multi-Event Probability
Handle 2, 3, or more events with AND/OR logic. The AI identifies whether events are independent or dependent and picks the right rule.
Compound & Geometric
Calculate the probability of repeated independent trials: first success on trial k (geometric), or combined outcomes across experiments.
Other calculators give you a number.
This one teaches you why.
Static probability tools (Omnicalculator, Calculator.net, Mathway) require you to already know the formula structure, the event type, and which form to fill in. They return a result with no context — useful for verification, useless for learning.
MathGPT Free identifies the problem type from your description first, then applies and explains each rule step by step. That diagnostic step is what most students get wrong — and no static form handles it.
The formulas the AI applies — and explains
Every probability type has a specific rule. The AI picks the right one, substitutes your values, and narrates each step.
From problem description
to answer in 4 steps
Describe your problem
Type your question in plain text — exact values like P(A) = 0.3, a word problem, or even just “10 coin flips, what’s the chance of 7 heads?” No special syntax needed.
Choose type — or let AI decide
Pick from the tabs (single, conditional, binomial, normal) or leave on Auto. The AI identifies the correct probability model from your description.
AI applies the right formula
The solver selects the rule, substitutes your values, and computes each arithmetic step with labeled annotations — not just the final answer.
Learn the reasoning, not just the result
Each step explains why — why we divide by P(B), when to use complement, what a z-table value represents. Understand the method so you can apply it yourself.
Built for every
probability situation
Stuck on a homework problem at 11pm
Paste the problem exactly as written — conditional, binomial, or multi-event — and get the formula applied with every step explained before submission.
Verifying your distribution setup
Check your z-score calculation, binomial cumulative formula, or Bayes’ theorem setup before submitting. See the AI’s method alongside yours to catch any errors.
Learning when each rule applies
Use worked examples to understand when conditional vs. compound probability applies, when the complement method is faster, and how to frame a problem algebraically.
Common questions
What Is a Probability Calculator — and When Do You Need One?
A probability calculator is a tool that computes the likelihood of specific outcomes occurring across different types of experiments — from drawing colored balls out of a bag to calculating the chance of passing a standardized test by guessing. The core idea is always the same: divide the number of favorable outcomes by the total number of equally likely outcomes. But in practice, problems quickly grow more complex than that single formula handles.
You need a probability calculator when the calculation involves multiple steps — conditional events where outcomes depend on each other, distributions where you need to integrate across a continuous range, or compound scenarios where three or four events interact. At that point, setting up the right formula matters as much as running the arithmetic, and that’s where an AI-powered calculator earns its keep: it identifies which formula applies, explains why, and walks through every substitution.
Single Event vs. Compound Probability: Which Formula Applies?
The simplest form of probability involves one event and one sample space. P(A) = favorable outcomes / total outcomes. Rolling a specific face on a fair die: 1/6. Drawing a red card from a standard deck: 26/52 = 0.5. These are classical probability problems, and the formula is straightforward.
Compound probability is where the rules branch. When events are independent — meaning one outcome doesn’t affect the other — the multiplication rule applies: P(A and B) = P(A) × P(B). When events are dependent, that formula no longer holds, and you need the general multiplication rule: P(A and B) = P(A) × P(B|A). The difference between these two setups is often where students make mistakes, and it’s what makes a probability calculator that explains its reasoning genuinely useful.
Before entering values, identify whether your events are independent or dependent. If drawing without replacement (cards, balls without putting them back), the events are dependent and P(B|A) ≠ P(B). That single distinction determines which formula the solver will use.
How to Use a Conditional Probability Calculator
Conditional probability answers the question: given that B already happened, what’s the probability of A? The formula is P(A|B) = P(A∩B) / P(B). This appears in Bayes’ theorem, disease testing scenarios, classroom grade problems, and any situation where partial information changes the likelihood of an outcome.
To use the conditional probability calculator here, you can either:
- Enter exact values — P(A) = 0.6, P(B) = 0.4, P(A and B) = 0.2 — and ask for P(A|B)
- Describe the scenario in words — “20 students: 12 passed math, 8 passed both math and science. P(science | math)?”
The AI extracts the relevant probabilities, applies the conditional formula, and verifies the result by checking consistency with the marginal probabilities provided.
Binomial Distribution Calculator: When to Use It
The binomial probability formula applies when you have a fixed number of independent trials, each with the same probability of success. The formula for exactly k successes in n trials is:
P(X = k) = C(n,k) × p^k × (1−p)^(n−k)
This shows up in coin-flip sequences, multiple-choice guessing scenarios, quality control sampling, and genetics problems. Most real problems ask for cumulative probability — “at most 3 successes” or “at least 6 correct” — which requires summing binomial terms. The calculator handles this in one step, showing the individual terms being summed.
Normal Distribution Probability and Z-Scores
When a population follows a normal distribution, probability is calculated as the area under the bell curve. The z-score standardizes any normal variable: z = (x − μ) / σ. Once you have a z-score, use our z-table calculator to instantly find the cumulative probability instead of manually reading a printed table.
- Below a value: P(X < x) — left-tail area
- Above a value: P(X > x) = 1 − P(X < x)
- Between two values: P(x₁ < X < x₂) = P(X < x₂) − P(X < x₁)
Multi-Event Probability: Independent and Dependent Events
When three or more events are involved, the structure matters more than the arithmetic. The probability of at least one event occurring is often easiest to find using the complement: P(at least one) = 1 − P(none). This avoids enumerating all combinations of which events occur.
For the intersection of independent events, multiply all individual probabilities. For unions, the inclusion-exclusion principle applies. Describe your setup in natural language and the AI will label each step of the calculation.
Don’t confuse geometric probability (first success on trial k) with binomial probability (exactly k successes in n trials). If your problem says “how many attempts until the first…”, that’s geometric. If it says “out of n attempts, how many…”, that’s binomial.
Why an AI Probability Calculator Beats a Static Form
Static probability calculators require you to already know the formula structure. They’re useful for verification, but they don’t help you understand whether you should use the binomial formula, or whether your problem is actually conditional, or whether the complement method would be simpler.
An AI-powered probability calculator handles the diagnostic step — identifying the problem type from your description — before computing anything. That’s the step students most often get wrong. The result is a tool that helps you learn the reasoning behind probability, not just run numbers through a formula you may be misapplying.
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