Binomial Cumulative Distribution Calculator

This calculator computes the cumulative probability for a binomial distribution: given the number of trials, the probability of success per trial, and a target number of successes, it returns the probability of observing that many or fewer successes. Enter the three values and you get the cumulative probability P(X ≤ k) instantly, along with the mean and standard deviation of the distribution.

Formula

The binomial distribution gives the probability of exactly k successes in n independent trials, each with success probability p. The probability mass function is:

P(X = k) = C(n, k) · p^k · (1 − p)^{n − k}

where C(n, k) is the binomial coefficient (n choose k), equal to n! / (k! · (n − k)!). The cumulative probability of observing k or fewer successes is the sum of that function from 0 to k:

P(X ≤ k) = Σ_i=0^k C(n, i) · pⁱ · (1 − p)^n−i

Variables:

• n – number of trials (nonnegative integer)
• p – probability of success on each trial (between 0 and 1)
• k – the upper bound for successes (integer between 0 and n)

Worked Example

Using the default values: n = 10 trials, p = 0.5, and k = 4. We want P(X ≤ 4).

Compute each term P(X = i) for i = 0, 1, 2, 3, 4:

• P(X=0) = C(10,0)·0.5⁰·0.5¹⁰ = 0.0009765625
• P(X=1) = C(10,1)·0.5¹Â·0.5⁹ = 0.009765625
• P(X=2) = C(10,2)·0.5²Â·0.5⁸ = 0.0439453125
• P(X=3) = C(10,3)·0.5³Â·0.5⁷ = 0.1171875
• P(X=4) = C(10,4)·0.5⁴·0.5⁶ = 0.205078125

Sum = 0.0009765625 + 0.009765625 + 0.0439453125 + 0.1171875 + 0.205078125 = 0.376953125.

The calculator shows 0.376953, matching our manual sum. The mean is np = 10·0.5 = 5, and the standard deviation is √(10·0.5·0.5) = √2.5 ≈ 1.58.

When to Use the Binomial Cumulative Distribution

Use this when you have a fixed number of independent trials, each with the same probability of success, and you want the probability of a range of outcomes. Common applications:

• Quality control: probability that at most 3 defective items appear in a batch of 50.
• Surveys: probability that fewer than 40% of 200 respondents favor a proposal.
• Games: probability of getting at most 2 heads in 10 coin flips.
• Medicine: probability that at most 5 patients out of 100 experience a side effect.

The cumulative version answers “at most k� or “fewer than k+1� questions. If you need “at least k�, use 1 − P(X ≤ k−1). For “between a and b�, subtract cumulative values.

How to Choose the Right Inputs

Three inputs are required:

• n (trials) – must be a positive integer. If your experiment has 100 trials, enter 100.
• p (probability) – a decimal between 0 and 1. For 30%, enter 0.3. Never enter 30.
• k (target) – an integer from 0 to n. The calculator sums probabilities from 0 up to k.

Make sure the trials are independent and the probability stays constant. If those conditions don’t hold, the binomial model isn’t appropriate. For example, drawing cards without replacement from a deck is not binomial because p changes after each draw.

Common Mistakes and Misconceptions

A frequent error is confusing “at most k� with “exactly k�. The cumulative distribution gives P(X ≤ k), not P(X = k). If you want exactly 4 successes, use the probability mass function instead (or compute the difference between cumulative values).

Another mistake is entering the probability as a percentage. Always convert to decimal. Also, check that k is not larger than n – the calculator will return an error or a meaningless result if k > n.

Some users mistakenly think the binomial distribution is symmetric. It is symmetric only when p = 0.5. For p ≠ 0.5, the distribution is skewed, and the cumulative probabilities reflect that skewness.

Limits and When the Formula Fails

The binomial distribution assumes:

• Fixed number of trials n.
• Each trial has exactly two outcomes (success/failure).
• Probability p is the same for every trial.
• Trials are independent.

If any of these are violated, the binomial model may give inaccurate probabilities. For large n (say, n > 100) and p not too close to 0 or 1, the normal approximation to the binomial is often used, but the exact cumulative calculation is still valid. For small n, the exact calculation is always preferred.

The calculator uses exact arithmetic with binomial coefficients, so it remains accurate even for large n, though very large n (thousands) may be slow on some devices.

Comparison with Other Distributions

The binomial distribution is closely related to other probability models:

• Bernoulli distribution – a single trial with two outcomes. The binomial is the sum of n independent Bernoulli trials.
• Poisson distribution – approximates the binomial when n is large and p is small (np stays moderate). It models rare events.
• Hypergeometric distribution – used when sampling without replacement from a finite population; probabilities change after each draw.
• Normal distribution – approximates the binomial for large n, especially when np and n(1−p) are both > 5.

Choosing the right distribution depends on the experiment. If trials are not independent, the binomial is not correct.

Frequently Asked Questions

What is the difference between binomial PDF and CDF?

The PDF (probability density function) gives the probability of exactly k successes. The CDF (cumulative distribution function) gives the probability of k or fewer successes. For example, P(X = 3) is the PDF, while P(X ≤ 3) is the CDF.

How do I calculate “at least 5 successes�?

Use the complement: P(X ≥ 5) = 1 − P(X ≤ 4). So compute the cumulative probability for k = 4 and subtract from 1.

Can I use this for percentages like 30%?

Yes, but convert the percentage to a decimal first. 30% becomes 0.3. Entering 30 would be interpreted as 3000%, which is invalid.

What if my probability of success is greater than 0.5?

That’s fine. The calculator works for any p between 0 and 1. Just enter the decimal, e.g., 0.8 for 80%.

Why is my result so small for large n?

Cumulative probabilities for extreme k values can be very small. For instance, with n=100 and p=0.5, the probability of 20 or fewer successes is minuscule because the distribution is centered around 50. The calculator shows the exact value, which may be in scientific notation.

Does the calculator work for non-integer k?

No, k must be an integer between 0 and n. The binomial distribution is defined only for integer counts of successes.

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in n | Number of trials (n) | - | 10 | 0
in p | Probability of success (p) | - | 0.5 | 0
in k | Target successes (k) | - | 4 | 0
out cum | P(X ≤ k) | - | sum(ncr(n,i)*p^i*(1-p)^(n-i), i, 0, k) | 6
out mean | Mean (np) | - | n*p | 2
out sd | Standard deviation | - | sqrt(n*p*(1-p)) | 2