Binomial Distribution Table Calculator & How to Use

This calculator computes binomial probabilities for a given number of trials (n), success probability (p), and the number of successes (k) you specify. You enter n, p, and the cutoff value k, then choose whether you want the probability of exactly k, at most k, or at least k successes. The result is the exact probability, along with the mean and standard deviation of the distribution, so you can use it instead of a printed binomial table.

Formula

The binomial probability formula gives the chance of exactly k successes in n independent trials, each with success probability p:

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

• n – number of trials (a positive integer)
• p – probability of success on a single trial (0 ≤ p ≤ 1)
• k – number of successes you are asking about (0 ≤ k ≤ n)
• C(n, k) – binomial coefficient, “n choose kâ€�, equal to n! / (k! · (n − k)!)

For “at most k� you add the probabilities for 0, 1, …, k. For “at least k� you add the probabilities for k, k+1, …, n. The mean is np and the standard deviation is √(np(1−p)).

Worked Example

Suppose you flip a fair coin 10 times (n = 10, p = 0.5) and want the probability of exactly 3 heads (k = 3).

1. Compute the binomial coefficient: C(10,3) = 10! / (3! · 7!) = 120.
2. Compute p^k = 0.5³ = 0.125.
3. Compute (1−p)^n−k = 0.5⁷ = 0.0078125.
4. Multiply: 120 × 0.125 × 0.0078125 = 0.1171875.

So P(X = 3) = 0.117188 (rounded to 6 decimals). The calculator shows 0.117188, the mean is 10×0.5 = 5, and the standard deviation is √(10×0.5×0.5) = 1.5811.

When to Use a Binomial Distribution

Use this distribution when your experiment meets all four conditions:

• There are a fixed number of trials (n).
• Each trial has only two outcomes: success or failure.
• The probability of success (p) is the same for every trial.
• The trials are independent – the outcome of one does not affect another.

Common examples include counting defective items in a batch, number of patients who respond to a treatment (with known response rate), number of free throws made in 10 attempts, or number of customers who click an ad. If any condition fails – for example, sampling without replacement from a small population – the binomial model may not be accurate; the hypergeometric distribution is then more appropriate.

How to Read the Result

The probability output is a number between 0 and 1. Multiply by 100 to get a percentage. For instance, 0.117188 means 11.72%. The mean tells you the average number of successes you would expect over many repetitions, and the standard deviation tells you how much the count typically varies. A larger n or a p closer to 0.5 makes the distribution more symmetric and closer to a normal curve.

If you select “at most k�, you get the cumulative probability P(X ≤ k). If you select “at least k�, you get P(X ≥ k). These are useful for hypothesis testing or quality control, where you often want the probability of seeing a result as extreme as the observed one.

Common Mistakes and Misconceptions

• Using p as a percentage: Enter p as a decimal, not a percentage. 25% is 0.25, not 25.
• Confusing “at mostâ€� with “exactlyâ€�: “At most 3â€� includes 0, 1, 2, and 3. “Exactly 3â€� is just one value. The difference can be huge.
• Forgetting independence: If trials are not independent (e.g., drawing cards without replacement), the binomial formula is wrong.
• Rounding too early: In hand calculations, keep several decimal places until the final step. Rounding p^k or the coefficient early can change the last digits.
• Using n and k interchangeably: n is the total trials, k is the number of successes you are asking about. Swapping them gives a different (often meaningless) result.

Limitations and Alternatives

The binomial distribution assumes a fixed n and constant p. In real life, p may drift, or trials may not be independent. When n is large (say > 50) and p is not too close to 0 or 1, the normal approximation is often used, but the exact binomial is still preferable when available. For rare events (small p) and large n, the Poisson distribution is a good approximation. If you are sampling without replacement from a finite population, use the hypergeometric distribution instead. This calculator gives exact probabilities, so it is suitable for most textbook and practical problems.

Frequently Asked Questions

What is a binomial distribution table?

A binomial distribution table lists cumulative probabilities P(X ≤ k) for various n and p values. Instead of calculating each term, you look up the row for your n and the column for your p, then read the cumulative probability for k. These tables typically cover only specific p values (like 0.05, 0.10, 0.25, 0.50) and limited n, so a calculator is more flexible.

How do I find P(X ≥ k) from a table?

Most tables give P(X ≤ k). To get P(X ≥ k), use the complement: P(X ≥ k) = 1 − P(X ≤ k−1). For example, if you want P(X ≥ 3) and the table gives P(X ≤ 2), subtract that value from 1. The calculator does this automatically when you choose “at least k�.

Can I use this for non-integer k?

No. The number of successes k must be a whole number between 0 and n. If you enter a decimal, the calculator will treat it as the floor (round down) or you should round to the nearest integer. In practice, you only ask for whole counts.

What if p is greater than 0.5?

That is fine. The formula works for any p between 0 and 1. For example, if a player makes 80% of free throws, p = 0.8. The distribution will be skewed toward higher counts of successes, and the mean np will be closer to n.

Why is my result different from a normal approximation?

The normal approximation is an approximation. The exact binomial probability is always correct for the model. Differences arise because the normal curve is continuous while the binomial is discrete. For small n or extreme p, the approximation can be off by several percent. Use the exact calculator when precision matters.

How do I know if my problem is binomial?

Check the four conditions: fixed n, two outcomes, constant p, independent trials. If any are violated, the binomial model is not appropriate. For example, if you draw 5 cards from a deck without replacement, the probability of a heart changes each draw – that is not binomial.

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in n | Number of trials (n) | - | 10 | 1
in p | Probability of success (p) | - | 0.5 | 0
in k | Number of successes (k) | - | 3 | 0
sel tail | Probability type | Exactly k=1, At most k=2, At least k=3
out prob | Probability | - | if(tail==1, ncr(n,k)*p^k*(1-p)^(n-k), if(tail==2, sum_{i=0}^{k} ncr(n,i)*p^i*(1-p)^(n-i), sum_{i=k}^{n} ncr(n,i)*p^i*(1-p)^(n-i))) | 6
out mean | Mean (np) | - | n*p | 2
out sd | Standard deviation | - | sqrt(n*p*(1-p)) | 2