Probability Calculator
with Step-by-Step Solutions

Probability Calculator with Steps — How It Works

A probability calculator with steps goes beyond returning a decimal answer. It shows which distribution applies to the problem, substitutes the given values into the formula, computes each intermediate calculation, and interprets the final result in plain English. This is useful both for checking homework answers and for understanding why a particular method was used.

Different probability problems require different approaches. A question about the number of heads in coin flips uses the binomial distribution. A question about waiting times or rare events uses Poisson. A question about standardized test scores or heights uses the normal distribution. Choosing the wrong distribution is one of the most common errors students make — this solver identifies the correct approach before calculating.

Probability Distributions Supported

Binomial Distribution

The binomial distribution models the number of successes in n independent Bernoulli trials, each with probability p. It requires three inputs: n (number of trials), k (desired number of successes), and p (success probability). The formula P(X=k) = C(n,k) × p^k × (1−p)^(n−k) is applied step by step, with the binomial coefficient C(n,k) calculated explicitly.

Normal Distribution

Normal distribution problems involve converting a raw score to a z-score and finding the corresponding area under the standard normal curve. The solver shows the z = (x − μ) / σ conversion, then looks up or calculates the cumulative probability for one-tailed or two-tailed problems.

Poisson Distribution

The Poisson distribution gives the probability of exactly k events occurring in a fixed interval when the average rate is λ. The formula P(X=k) = (e^−λ × λ^k) / k! is applied with all components calculated and shown.

Conditional Probability and Bayes’ Theorem

Conditional probability problems ask for P(A|B) — the probability of A given that B has occurred. The solver applies P(A|B) = P(A∩B) / P(B) and, for Bayes’ problems, works through the full posterior calculation with all prior and likelihood values shown.

Tips for Accurate Results

• Identify whether the problem gives you exact counts (binomial/Poisson) or continuous measurements (normal)
• For normal distribution problems, check whether you need the area to the left, right, or between two values
• For binomial problems, confirm that trials are independent and p is constant across trials
• For conditional probability, check whether events are mutually exclusive or independent before calculating