How do I read a binomial distribution table?

Pick the block for your number of trials n, go down to the row for k successes and across to the column for your success probability p. For n = 10, k = 3 and p = 0.30, the individual table gives 0.2668 and the cumulative table gives 0.6496.

What is the difference between individual and cumulative binomial tables?

An individual table gives P(X = k), the probability of exactly k successes. A cumulative table gives P(X ≤ k), the probability of k or fewer, which is the running total of the individual entries from 0 to k. Use cumulative for “at most” questions and ranges.

How do I find “at least” or “more than” probabilities?

Use the complement. P(X ≥ k) = 1 − P(X ≤ k − 1) and P(X > k) = 1 − P(X ≤ k). For n = 10 and p = 0.30, P(X ≥ 4) = 1 − 0.6496 = 0.3504. The calculator shows all five probabilities at once.

What do I do when p is greater than 0.5?

Swap successes and failures. P(X = k; n, p) = P(X = n − k; n, 1 − p), so for n = 10, p = 0.70 and k = 7, look up k = 3 with p = 0.30 and read 0.2668. For cumulative values, P(X ≤ k; n, p) = 1 − P(X ≤ n − k − 1; n, 1 − p).

What if my n or p is not in the table?

The tables cover n = 1 to 20 and 25 with p from 0.01 to 0.50. For n = 21 to 24, anything above 25 up to 1000 and any p from 0 to 1, use the calculator above. For example, n = 1000, p = 0.30 and k = 300 gives P(X = 300) = 0.0275.

Why does my textbook or software show a different last digit?

Entries here are rounded half up to four decimals from exact values, so 0.03125 appears as 0.0313. Some software rounds binary floating-point numbers instead and prints 0.0312 for the same value. Older printed tables may also differ in the last digit, so compare to about three decimals.

Binomial Distribution Table

Binomial distribution table for n = 1 to 20 and 25: individual P(X = k) and cumulative P(X ≤ k), plus a calculator up to n = 1000. Printable.

Individual P(X = k) and cumulative P(X ≤ k)Calculator for n up to 1000, any pPNG, SVG and print on A4 or Letter, no sign-up

Binomial probability calculator

n (trials)

p (success)

k (successes)

Examples:

Plot these probabilities as a bar chart

Copy the k, P(X = k) rows for your n and p, then paste them into the bar chart maker as Label, value rows. Its value labels show two decimals.

Open bar chart maker

P(X = 5)

0.2461

P(X ≤ 5)

0.6230

P(X ≥ 5)

0.6230

P(X < 5)

0.3770

P(X > 5)

0.3770

Mean np

Variance np(1 − p)

2.5

Standard deviation

1.58114

Chart style

Binomial probability tables, n = 1 to 20 and 25

Choose n, then read down to k and across to p. The first tab gives the probability of exactly k successes, the second the probability of k or fewer. For p above 0.5, use the symmetry rule under the table.

Number of trials n

Export style

Paper for print

Binomial probabilities P(X = k) for n = 10, by k and p
k \ p	0.01	0.05	0.10	0.15	0.20	0.25	0.30	0.35	0.40	0.45	0.50
0
1
2
3
4
5
6
7
8
9
10

0.0000 means the probability is below 0.00005. It is not exactly zero.

p above 0.5: swap successes and failures. P(X = k; n, p) = P(X = n − k; n, 1 − p), so P(X = 7; n = 10, p = 0.70) is the table entry for k = 3, p = 0.30. For cumulative values, P(X ≤ k; n, p) = 1 − P(X ≤ n − k − 1; n, 1 − p).

Values are rounded half up to four decimals from exact values. We compared every cell with SciPy (binom.pmf and binom.cdf); the only differences are cells exactly on a rounding boundary, such as 0.03125. Click any cell to see it spelled out.

Selected cell

n = 10, p = 0.30, P(X = 3) = 0.2668

More digits: 0.2668.

Free Binomial Distribution Table Charts

Every chart below is drawn from the same code as the tool above. View, download and print them free, no sign-up.

Individual

Binomial Distribution Table: P(X = k), n = 10

Exactly-k probabilities for ten trials. For k = 5 and p = 0.50 the entry is 0.2461.

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Reference tables

How to read

Distribution

Printable versions

Moving from counts to continuous data? Look up normal areas in the z table, critical values in the t table or the chi-square table, and draw a probability bar chart with the bar chart maker.

What a binomial distribution table shows

A binomial distribution table lists probabilities for a count of successes. Use it when you repeat the same trial n times and count how many times a result happens: heads in 10 coin flips, defective parts in a batch of 20, correct answers on a multiple-choice quiz.
The binomial model fits when four conditions hold. There is a fixed number of trials n. Each trial has exactly two outcomes, called success and failure. The probability of success p is the same on every trial. The trials are independent. If you draw without replacement from a small group, the trials are not independent and the binomial is only an approximation.
The probability of exactly k successes is P(X = k) = C(n, k) p^k (1 − p)^(n − k), where C(n, k) = n! ÷ (k!(n − k)!). The mean is np and the variance is np(1 − p).

How to read the table

Choose the block for your n, go down to k and across to p. In the individual tab, n = 10, k = 3 and p = 0.30 gives 0.2668, so the chance of exactly 3 successes in 10 trials is 0.2668. In the cumulative tab the same position gives 0.6496, the chance of 3 or fewer. That entry is the sum of the first four individual entries: 0.0282 + 0.1211 + 0.2335 + 0.2668 = 0.6496. The cumulative entry is the exact sum rounded once, so adding rounded entries can occasionally differ in the last digit. Some textbooks print P(X ≥ k) instead of P(X ≤ k), so check the column heading before you read a value.
A zero in the table is rounded, not exact. With n = 20 and p = 0.05, the chance of no successes is 0.3585, while P(X = 20) shows 0.0000 because it is smaller than 0.00005.

At least, more than, fewer than

The cumulative tab only answers “k or fewer”, but the other questions are one subtraction away. P(X ≥ k) = 1 − P(X ≤ k − 1). P(X > k) = 1 − P(X ≤ k). P(X < k) = P(X ≤ k − 1). For n = 10 and p = 0.30, the chance of at least 4 successes is 1 − 0.6496 = 0.3504. The calculator at the top of the page gives all five probabilities, P(X = k), P(X ≤ k), P(X ≥ k), P(X < k) and P(X > k), in one go.

When p is greater than 0.5

The table stops at p = 0.50 because of symmetry. Counting successes with probability p is the same as counting failures with probability 1 − p, so P(X = k; n, p) = P(X = n − k; n, 1 − p). To find P(X = 7) for n = 10 and p = 0.70, look up k = 3 with p = 0.30: 0.2668.
For cumulative values use P(X ≤ k; n, p) = 1 − P(X ≤ n − k − 1; n, 1 − p). For n = 10, p = 0.70 and k = 7, that is 1 − P(X ≤ 2; n = 10, p = 0.30) = 1 − 0.3828 = 0.6172. If p is not in the table, or n is 21 to 24 or above 25, use the calculator, which accepts any p between 0 and 1 and n up to 1000.

Normal approximation: when it works

For large n the binomial can be approximated by a normal distribution with mean np and standard deviation √(np(1 − p)). A common rule of thumb asks for np ≥ 10 and n(1 − p) ≥ 10; some textbooks use 5 instead of 10. It is a guideline, not a guarantee.
Example: n = 100 and p = 0.4 gives np = 40 and n(1 − p) = 60, so the rule is met. The exact P(X ≤ 45) is 0.8689. The normal approximation with a continuity correction, Φ((45.5 − 40) ÷ √24), gives 0.8692. Without the correction, Φ((45 − 40) ÷ √24) gives 0.8463, which is further off. For a z-score lookup, use the z table.

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