What Is Binomial Distribution With Steps?
The binomial distribution counts successes in n independent trials with the same success probability p. Each term is C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ, and range probabilities add several terms.
Enter n and p, choose exact, at most, at least or between, and the tool lists each term with its binomial coefficient and a running total. It also gives the mean, variance, standard deviation and the normal approximation with continuity correction.
The sections below cover the formulas, a worked range example and when the normal approximation is accurate.
How to Use the Binomial Distribution With Steps Calculator
- Enter n and p.
- Choose exact, at most, at least or between.
- Enter k.
- Read each term and the total.
Binomial Distribution With Steps Formula
Binomial probability
Range probability
Binomial Distribution With Steps Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| n | Number of trials | |
| p | Probability of success | |
| k | Number of successes | |
| C(n, k) | Binomial coefficient |
Binomial Distribution With Steps Worked Example
With n = 12 and p = 0.3, P(2 ≤ X ≤ 5) adds the terms for k = 2, 3, 4 and 5: 0.1678 + 0.2397 + 0.2311 + 0.1585 = 0.7971 (rounded terms).
With the example values loaded in the calculator above, the steps are:
- Each term is P(X = k) = C(12, k) · 0.3^k · 0.7^(12 − k).
- k = 2: C(12, 2) = 66; term = 0.1677903
- k = 3: C(12, 3) = 220; term = 0.23970043
- k = 4: C(12, 4) = 495; term = 0.2311397
- k = 5: C(12, 5) = 792; term = 0.15849579
- P(2 ≤ X ≤ 5) = sum of the terms = 0.79712621.
Result: P(2 ≤ X ≤ 5) = 0.79712621.
How to Read a Binomial Range Probability
A binomial counts successes in n independent trials with the same chance p each time. With n = 12 and p = 0.3, the mean is 3.6 and the variance is 2.52. The chance of between 2 and 5 successes is 0.1678 + 0.2397 + 0.2311 + 0.1585 = 0.7971.
The term-by-term table shows every probability that was added, so you can follow each step. The normal approximation with a continuity correction gives a close value when np and n(1 − p) are both large enough.
Where Binomial Distribution With Steps Is Used
Range probabilities
Add the terms for k₁ to k₂ and see each one.
At least or at most
Find P(X ≥ k) or P(X ≤ k) for quality or reliability problems.
Mean and spread
Read the mean np and the variance np(1 − p) of the count.
Checking an approximation
Compare the exact answer with the normal approximation.
Common Binomial Distribution With Steps Mistakes to Avoid
- The trials must be independent with the same p each time.
- To find "at least k" add the terms from k to n, or take 1 minus P(X ≤ k − 1).
- Use a continuity correction when you approximate a binomial with a normal curve.
- Rounded terms add up to a slightly different total than the exact value. The calculator uses the exact one.
Binomial Distribution With Steps Terms and Definitions
- Trial
- One repetition with two outcomes, success or failure.
Binomial Distribution With Steps FAQs
How do you calculate a binomial range probability?
Add the individual probabilities P(X = k) for each k in the range. Each term is C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ.
What are the mean and variance?
The mean is np and the variance is np(1 − p). For n = 12 and p = 0.3 these are 3.6 and 2.52.
When can I use the normal approximation?
A common rule is that both np and n(1 − p) are at least about 10, though some texts use 5. The approximation improves as n grows.
How is at least k found?
Add P(X = k) through P(X = n), or compute 1 − P(X ≤ k − 1). The second way is shorter when k is small.