Binomial Distribution Calculator
Calculate binomial probabilities for a fixed number of trials, success probability, and number of successes. Use this student-friendly calculator for exact, less-than, greater-than, between-values, and cumulative binomial distribution problems with a shaded bar chart, formulas, probability table, and step-by-step explanations.
Background
The binomial distribution models the number of successes in a fixed number of independent trials when each trial has only two outcomes: success or failure. It is commonly used for quiz guessing, survey responses, product defects, free throws, clinical outcomes, and other repeated yes/no situations.
How to use this calculator
- Enter the number of trials n.
- Enter the probability of success p.
- Choose whether you want an exact, cumulative, between-values, or outside-values probability.
- Enter the number of successes x, or a lower and upper value for range questions.
- Click Calculate to see the probability, bar chart, formula, statistics, table, and steps.
How this calculator works
- It uses the binomial probability formula to calculate the probability of each possible number of successes.
- For cumulative questions, it adds the probabilities of all included outcomes.
- It highlights the included bars on the probability chart.
- It reports the expected value, variance, and standard deviation of the binomial distribution.
- It explains the result in plain language so students understand what the probability means.
Formula & Equations Used
Binomial probability: P(X = k) = C(n,k)p^k(1-p)^(n-k)
Combination: C(n,k) = n! / [k!(n-k)!]
Expected value: E(X) = np
Variance: Var(X) = np(1-p)
Standard deviation: σ = √[np(1-p)]
Cumulative probability: add all included P(X = k) values.
Example Problem & Step-by-Step Solution
Example 1 — Find P(X = 7)
- A quiz has 10 true/false questions.
- A student guesses randomly, so p = 0.5.
- We want the probability of exactly 7 correct answers.
- Use n = 10, p = 0.5, and x = 7.
- Apply P(X = k) = C(n,k)p^k(1-p)^(n-k).
- The calculator highlights the bar for X = 7 and reports the probability.
Example 2 — Find P(X ≥ 7)
- Use the same quiz setup: n = 10, p = 0.5.
- Choose P(X ≥ x).
- Enter x = 7.
- The calculator adds P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10).
- The graph highlights all bars from 7 through 10.
Example 3 — Find P(6 ≤ X ≤ 9)
- A basketball player has a free-throw success probability of p = 0.75.
- The player shoots n = 10 free throws.
- Choose between-values mode.
- Enter lower value 6 and upper value 9.
- The calculator adds the probabilities for 6, 7, 8, and 9 successes.
Frequently Asked Questions
Q: What is a binomial distribution?
A binomial distribution gives the probability of getting a certain number of successes in a fixed number of independent success/failure trials.
Q: What does n mean in a binomial distribution?
The value n is the number of trials, such as 10 quiz questions or 20 surveyed people.
Q: What does p mean in a binomial distribution?
The value p is the probability of success on one trial.
Q: What is the difference between exact and cumulative binomial probability?
Exact probability finds one outcome, such as P(X = 7). Cumulative probability adds multiple outcomes, such as P(X ≤ 7) or P(X ≥ 7).
Q: Can this calculator show the probability table?
Yes. It can show a table of P(X = k) values for every possible number of successes from 0 to n.
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