What is the "At Least k Successes" Binomial Calculator?

This tool computes the probability of obtaining at least k successes in n independent trials, where each trial succeeds with the same probability p. This is the upper-tail cumulative probability of the binomial distribution, written $P(X \geq k)$. It is widely used in quality control, reliability testing, A/B testing, polling, and any scenario built from repeated yes/no (Bernoulli) trials.

How to Use It

Enter three values: the number of trials n (a whole number), the minimum number of successes k you care about, and the per-trial success probability p as a decimal between 0 and 1 (for example 0.25 for 25%). Press calculate to see $P(X \geq k)$, the same value as a percentage, plus the exact-k probability $P(X = k)$ and the lower tail $P(X \leq k)$.

The Formula Explained

The probability of exactly i successes is the binomial mass function $C(n, i) \cdot p^{i} \cdot (1-p)^{n-i}$, where $C(n, i)$ is the number of ways to choose which trials succeed. To get "at least k," we sum every term from $i = k$ up to $i = n$:

$$P(X \geq k) = \sum_{i=k}^{n} \binom{n}{i}\, p^{\,i}\,\left(1-p\right)^{n-i}$$

The calculator builds each term iteratively for numerical stability rather than computing large factorials directly, so it stays accurate even for large n.

Bar chart of a binomial distribution with bars from k to n shaded to show the at-least-k tail probability — $P(X \geq k)$ is the sum of the shaded upper-tail bars of the binomial distribution.

Worked Example

Suppose you flip a fair coin ($p = 0.5$) 10 times and want the chance of getting at least 6 heads. With $n = 10$, $k = 6$, $p = 0.5$, the sum of terms for $i = 6, 7, 8, 9, 10$ equals $$\frac{386}{1024} \approx 0.376953,$$ or about 37.7%.

Diagram showing two coins per trial with success and failure paths and combinatorial selection of successes — Each successful trial contributes a factor of $p$ and each failure a factor of $(1-p)$, summed over all ways to choose $i$ successes.

Interpreting Your Result

$P(X \geq k)$ is a one-sided (upper-tail) probability: it answers "if the true success rate really is $p$, how often would I see $k$ or more successes in $n$ trials, just by chance?" It bundles together every outcome from exactly $k$ successes up through all $n$.

A small result — for example below 0.05 — means the observed count of successes would be surprising under the assumed $p$. That is exactly the logic behind a one-sided p-value: if you assume a baseline rate and your data lands far in the tail, the assumption looks doubtful. A large result means the count is unremarkable and entirely consistent with the assumed $p$.

A/B testing. If the control conversion rate is $p$ and the variant produced $k$ of $n$ conversions, $P(X \geq k)$ gauges whether the lift could just be noise. A tiny tail probability is evidence the variant genuinely differs.
Quality control / acceptance sampling. With an assumed defect rate $p$, $P(X \geq k)$ is the chance a lot shows $k$ or more defectives in a sample of $n$ — the foundation of accept/reject rules.
Reliability "at least one". Setting $k=1$ gives the probability that at least one event occurs across $n$ independent attempts.

For large $n$, the binomial tail is often approximated by a normal distribution, so a normal upper-tail tool can serve as a sanity check once $np$ and $n(1-p)$ are both comfortably above about 10. Treat the number as a description of how compatible your data is with the assumed $p$; choosing a threshold for action is a study-design decision, not something the probability dictates on its own.

Definitions & Glossary

$n$ — number of trials: The fixed total count of independent repetitions of the experiment (e.g. 20 coin flips, 100 sampled parts).
$k$ — minimum number of successes: The threshold being tested. $P(X \geq k)$ sums the probabilities of getting exactly $k, k+1, \dots, n$ successes.
$p$ — per-trial success probability: The probability that any single trial is a "success," assumed identical for every trial. It lies between 0 and 1.
Bernoulli trial: A single experiment with exactly two outcomes — success (probability $p$) or failure (probability $1-p$). A binomial setting is $n$ identical, independent Bernoulli trials.
Binomial coefficient $\binom{n}{i}$: "$n$ choose $i$," the number of distinct ways to arrange $i$ successes among $n$ trials: $\binom{n}{i} = \dfrac{n!}{i!\,(n-i)!}$.
Cumulative / upper-tail probability: A probability that aggregates a range of outcomes. $P(X \geq k)$ is the upper tail — it adds the chances of all counts from $k$ up to $n$. Its complement is $P(X \leq k-1)$.
Independence: The assumption that the outcome of one trial does not influence any other. Without independence (and a constant $p$), the binomial formula does not apply.

FAQ

Does this assume independent trials? Yes. Each trial must be independent and have the same success probability $p$.

What if I want exactly k or at most k? The result table also shows $P(X = k)$ and $P(X \leq k)$ for convenience.

Can p be a percentage? Enter p as a decimal (e.g. 0.05 for 5%), not as 5.

P(X = k) exactly k successes	0.117188
P(X ≤ k) at most k successes	0.171875

Probability of At Least k Successes (Binomial)

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