Binomial Distribution Calculator With Steps

Q: How do I calculate binomial probabilities on a TI-84?

Press 2nd then VARS (DISTR). Use binompdf(n, p, x) for P(X = x) and binomcdf(n, p, x) for P(X ≤ x). For “at least x”, calculate 1 − binomcdf(n, p, x − 1). For the quiz example, 1 − binomcdf(10, 0.25, 5) = 0.0197.

Q: How do I calculate binomial probabilities in Excel or Google Sheets?

Use =BINOM.DIST(x, n, p, FALSE) for P(X = x) and =BINOM.DIST(x, n, p, TRUE) for P(X ≤ x). For a range, =BINOM.DIST.RANGE(n, p, a, b) gives P(a ≤ X ≤ b). Google Sheets uses the same functions (and the older BINOMDIST).

How to use the binomial distribution calculator

Enter the number of trials (n) and the probability of success on each one (p), as a decimal or a percentage.
Choose the question: exactly, at most, fewer than, at least, more than or between.
Enter the number of successes (x), or a and b for a range, and click Calculate probability.
Read the probability and the plain-English summary. The chart highlights the outcomes you asked about.
Check the steps and the full table of probabilities, which also show the mean and standard deviation.

Binomial probability formula

The probability of exactly k successes in n independent trials, each with success probability p:

P(X = k) = (nk) p^k(1 − p)^{n − k} , (nk) = n!k!(n − k)!

“At most”, “at least” and “between” add up these probabilities for every k in the range. The mean and standard deviation are:

μ = np , σ = np(1 − p)

n: The number of trials
p: The probability of success on each trial (and 1 − p of failure)
k: A number of successes, from 0 to n
(nk): “n choose k”: the number of ways to pick which k of the n trials are the successes
μ, σ: The mean (expected number of successes) and standard deviation of the number of successes

Worked example

A basketball player makes 70% of her free throws. What is the probability she makes exactly 3 of her next 5?

Identify n, p and k
n = 5, p = 0.7, k = 3
Count the ways to choose which 3 shots go in
(53) = 5!3! 2! = 10
Multiply by the probability of any one such sequence
P(X = 3) = 10 × 0.7³ × 0.3² = 10 × 0.343 × 0.09 = 0.3087

What it means: she makes exactly 3 of 5 about 31% of the time. Her expected number is μ = 5 × 0.7 = 3.5, so 3 or 4 makes are the most common outcomes.

How to interpret the result

The probability is the long-run share of repetitions in which the event happens. 0.02 means about 2 times in 100.
Unusual results: a probability below 0.05 for “at least” or “at most” a value suggests the result would be surprising if p were really what you assumed. That's the idea behind a hypothesis test.
The mean and standard deviation tell you what's typical: most outcomes fall within about two standard deviations of np.
The chart shows the shape: symmetric when p = 0.5, skewed right when p is small and skewed left when p is large.

For large n, the binomial looks like a bell curve; compare with the normal distribution calculator. The z-score calculator helps place a count relative to the mean.

Frequently asked questions

When can I use the binomial distribution?

When four conditions hold (often remembered as BINS): Binary outcomes (each trial is a success or a failure), Independent trials, a fixed Number of trials, and the Same probability of success on every trial. Coin flips, guessing on multiple-choice questions, defective items in a batch and free throws usually qualify. Drawing cards without putting them back does not, strictly, because the probability changes after each draw.

What is the difference between “at most” and “fewer than”?

“At most 3” means 3 or fewer: P(X ≤ 3), which includes 3. “Fewer than 3” means 0, 1 or 2: P(X < 3) = P(X ≤ 2). Likewise “at least 3” is P(X ≥ 3) and “more than 3” is P(X > 3) = P(X ≥ 4). Because the binomial counts whole numbers, including or excluding the endpoint changes the answer.

How do I calculate binomial probabilities on a TI-84?

Press 2nd then VARS (DISTR). Use binompdf(n, p, x) for P(X = x) and binomcdf(n, p, x) for P(X ≤ x). For “at least x”, calculate 1 − binomcdf(n, p, x − 1). For the quiz example, 1 − binomcdf(10, 0.25, 5) = 0.0197.

How do I calculate binomial probabilities in Excel or Google Sheets?

Use =BINOM.DIST(x, n, p, FALSE) for P(X = x) and =BINOM.DIST(x, n, p, TRUE) for P(X ≤ x). For a range, =BINOM.DIST.RANGE(n, p, a, b) gives P(a ≤ X ≤ b). Google Sheets uses the same functions (and the older BINOMDIST).

When can I use the normal approximation to the binomial?

A common rule is when both np ≥ 10 and n(1 − p) ≥ 10 (some books use 5). Then the binomial is close to a normal distribution with mean np and standard deviation √(np(1 − p)), with a continuity correction of ±0.5. With a calculator like this one you don’t need the approximation, but the result shows whether it would be reasonable.

What are the mean and standard deviation of a binomial distribution?

The mean (expected number of successes) is μ = np, the variance is σ² = np(1 − p) and the standard deviation is σ = √(np(1 − p)). For 10 guessed questions with p = 0.25, you expect 2.5 right on average, give or take about 1.37.