Understanding your binomial-distribution result
The table below summarizes how each calculation mode is defined and when it is the appropriate choice.
| Mode | Question answered | Formula |
|---|---|---|
| Exactly k | What is the probability of exactly k successes? | P(X = k) |
| At most k | What is the probability of k or fewer successes? | P(X ≤ k) |
| At least k | What is the probability of k or more successes? | P(X ≥ k) |
- A commonly cited rule of thumb (found in standard statistics texts) is that the binomial distribution can be reasonably approximated by a normal distribution when both n × p and n × (1 − p) are at least 5, with a continuity correction improving the approximation further.
- The binomial model requires independent trials with a constant success probability; sampling without replacement from a small finite population violates independence and calls for the hypergeometric distribution instead.
- k must be a whole number between 0 and n inclusive; non-integer or out-of-range values of k are not valid binomial outcomes.
- "At most k" and "at least k + 1" are complementary and sum to 100% (or, equivalently, "at most k − 1" and "at least k" sum to 100%).
What is the binomial distribution?
The binomial distribution models the number of successes in a fixed number of independent trials (n), each with the same probability of success (p) and only two possible outcomes per trial — success or failure. Classic examples include the number of heads in a fixed number of coin flips, or the number of defective items in a fixed sample from a production line, provided the success probability stays constant across trials.
The distribution is discrete: the number of successes k can only take whole-number values from 0 to n. Its shape depends on both n and p — symmetric when p = 0.5, and increasingly skewed as p moves toward 0 or 1.
The two defining assumptions are independence (the outcome of one trial does not affect another) and a constant success probability across all trials. When trials are drawn without replacement from a small, finite population, these assumptions can be violated and the related hypergeometric distribution is more appropriate.
How to use this binomial distribution calculator
- Enter the number of trials, n.
- Enter the probability of success per trial, p, as a percentage.
- Enter the number of successes, k (a whole number from 0 to n).
- Choose a mode: exactly k successes, at most k successes, or at least k successes, and read the resulting probability, along with the expected value and standard deviation of the distribution.
The formula behind binomial probability
The probability of exactly k successes in n trials is given by the binomial probability mass function: P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, where C(n, k) = n! / (k!(n − k)!) is the number of ways to choose k successes among n trials.
"At most k" sums this probability for every outcome from 0 to k; "at least k" sums it from k to n (equivalently, 1 minus the probability of at most k − 1).
The distribution's expected value (mean number of successes) is n × p, and its standard deviation is √(n × p × (1 − p)).
Worked example: n = 20 trials, p = 50%, exactly k = 8 successes. P(X = 8) = C(20, 8) × 0.5⁸ × 0.5¹² ≈ 12.01%. The expected value is 20 × 0.5 = 10, and the standard deviation is √(20 × 0.5 × 0.5) = √5 ≈ 2.2361.
Common mistakes
- Using the binomial distribution for trials that are not independent or where the success probability changes between trials, such as drawing without replacement from a small population.
- Off-by-one errors between modes — "at least 4" is not the complement of "at most 4"; the complement of "at most 4" is "at least 5".
- Entering the success probability as a decimal (0.5) instead of a percentage (50) where the input expects a percentage.
- Applying the binomial distribution to a continuous outcome, or to a count with more than two possible results per trial (which instead needs a multinomial model).
- Forgetting that the binomial distribution assumes p stays constant across all n trials — a probability that drifts over time invalidates the model.
Perguntas frequentes
How do I calculate a binomial probability?
Use P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, where n is the number of trials, p is the success probability, and C(n, k) counts the number of ways to arrange k successes among n trials. For "at most" or "at least", sum this formula over the relevant range of k values.
What conditions must be met to use the binomial distribution?
There must be a fixed number of trials (n), each trial must have only two possible outcomes (success or failure), the probability of success (p) must stay constant across trials, and the trials must be independent of one another.
What is the expected number of successes?
The expected value (mean) of a binomial distribution is n × p. For example, with 20 trials and a 50% success probability, the expected number of successes is 20 × 0.5 = 10.
What is the difference between 'exactly', 'at most' and 'at least'?
"Exactly k" is the probability of precisely k successes. "At most k" sums the probabilities for 0 through k successes. "At least k" sums the probabilities for k through n successes; it equals 1 minus the probability of at most k − 1 successes.
When can a normal distribution approximate a binomial distribution?
A commonly used rule of thumb is that the approximation is reasonable when both n × p and n × (1 − p) are at least 5. Outside that range, particularly for small n or p far from 0.5, the exact binomial calculation is more reliable.
Referências
- National Institute of Standards and Technology (NIST). NIST/SEMATECH e-Handbook of Statistical Methods — binomial distribution. nist.gov.
- Moore DS, McCabe GP, Craig BA. Introduction to the Practice of Statistics. W. H. Freeman (binomial setting and probability formula).
- Casella G, Berger RL. Statistical Inference (2nd ed). Duxbury, 2002 (binomial distribution, mean and variance).
- Ross SM. A First Course in Probability. Pearson (binomial random variables and the normal approximation).