A binomial probability distribution calculator finds the chance of getting exactly x successes in n trials. It also finds the cumulative probabilities P(X < x), P(X ≤ x), P(X > x), and P(X ≥ x). This tool gives the mean, variance, and standard deviation too, with a chart and a full table.
What Is a Binomial Probability Distribution Calculator?
A binomial probability distribution calculator is a tool for problems with two outcomes, such as yes or no, heads or tails, or pass or fail. You call one outcome a “success.” The tool tells you how likely it is to get a certain number of successes.
For example, it can answer questions like these:
- What is the chance of getting exactly 5 heads in 10 coin flips?
- What is the chance that at least 2 of 20 products are faulty?
- What is the chance of rolling at least one six in 5 rolls of a die?
How to Use the Binomial Probability Distribution Calculator
- Enter the number of trials (n). This is how many times you repeat the test, like 10 coin flips.
- Enter the probability of success (p). Use a decimal from 0 to 1. Enter 0.5 for 50% or 0.3 for 30%.
- Enter the number of successes (x). This is the count you want the chance for.
- Press Calculate.
You will see all five probabilities at once. You will also see the mean, variance, standard deviation, a bar chart, and a table of every value.
When Is a Problem Binomial?
A problem is binomial only if all four of these are true:
- Fixed trials: You know the number of trials, n, before you start.
- Two outcomes: Each trial is a success or a failure.
- Same chance: The chance of success, p, is the same every time.
- Independent trials: One result does not change the next.
If you draw cards without putting them back, the chance changes each time. That is not binomial.
The Binomial Probability Formula
P(X = x) = C(n, x) × px × (1 − p)n − x
- C(n, x) is the number of ways to pick which x trials are successes. It equals n! ÷ (x! × (n − x)!).
- px is the chance of x successes.
- (1 − p)n − x is the chance of the rest being failures.
Mean and Standard Deviation
- Mean: μ = np. This is the number of successes you expect on average.
- Variance: σ² = np(1 − p)
- Standard deviation: σ = √(np(1 − p)). This shows how much results usually spread from the mean.
Which Probability Should You Use?
Word problems use different phrases. Here is how they match the results:
| Phrase in the question | Use |
|---|---|
| Exactly x | P(X = x) |
| Fewer than x, less than x | P(X < x) |
| At most x, no more than x | P(X ≤ x) |
| More than x, greater than x | P(X > x) |
| At least x, no fewer than x | P(X ≥ x) |
Worked Examples
Example 1: Coin Flips
You flip a fair coin 10 times. What is the chance of exactly 5 heads?
- n = 10, p = 0.5, x = 5
- C(10, 5) = 252
- P(X = 5) = 252 × 0.55 × 0.55 = 252 ÷ 1,024 ≈ 0.2461
So there is about a 24.6% chance. The chance of 5 heads or fewer, P(X ≤ 5), is about 0.6230.
Example 2: Faulty Products
A machine makes faulty parts 10% of the time. You check 20 parts. What is the chance of at least 2 faulty parts?
- n = 20, p = 0.1, x = 2
- P(X ≥ 2) ≈ 0.6083
- P(X = 0) ≈ 0.1216, so there is about a 12% chance of no faulty parts at all.
- Mean = 20 × 0.1 = 2 faulty parts
- Standard deviation ≈ 1.34
Example 3: Rolling a Six
You roll a die 5 times. What is the chance of at least one six?
- n = 5, p = 1/6 ≈ 0.1667, x = 1
- P(X ≥ 1) ≈ 0.5981
That is about a 60% chance. Enter p as a decimal, like 0.166667.
How to Read the Chart
The bar chart shows the chance of each possible number of successes. Your x is shown in a darker color. The tallest bars sit near the mean. When n is large, the chart zooms in on the values near the mean, because almost all of the probability is there.
Binomial Probability Distribution Calculator FAQs
Is This Tool Free?
Yes. It is free, and you do not need to sign up. It runs in your browser.
How Large Can n Be?
You can enter up to 10,000 trials. The tool uses a method that stays accurate for large numbers.
Can I Use Percentages?
Enter p as a decimal. For 25%, enter 0.25. For 1 in 6, enter about 0.1667.
What Is the Difference Between Binomial PDF and CDF?
The PDF (probability mass) is the chance of exactly x, or P(X = x). The CDF is the chance of x or fewer, or P(X ≤ x). The tool shows both, so you do not need two separate tools.
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Final Thoughts
A binomial probability distribution calculator gives you exact and cumulative probabilities in one step. It also shows the mean, standard deviation, and the full shape of the distribution. Use it for homework, quality checks, games of chance, or any problem with a fixed number of yes-or-no trials.