Binomial Distribution Table

What Is a Binomial Distribution Table?

A binomial distribution table lists the probability of every possible number of successes, x = 0, 1, 2 and so on up to n, for a fixed number of trials n and one or more success probabilities p. Each cell is either the exact probability P(X = x) or the cumulative probability P(X ≤ x), depending on how the table was printed.

Printed tables were the standard tool before calculators became cheap. A textbook appendix usually gives one small table for each n from 2 to about 20, with x down the side and p across the top in steps of 0.05 or 0.10. This page builds that same table on demand, for any n you type, with the p values you choose. It also does the part a printed page cannot: it gives you the exact cell you asked for, the neighbouring tail probabilities, and a chart of the whole column.

If you only need one probability and not the full grid, the binomial probability calculator is quicker. If you need running totals or the smallest k for a target probability, use the binomial CDF calculator. This page is for the cases where you want to see the whole table, check a printed one, or copy it into a document.

How Do You Use This Binomial Table Calculator?

Enter n, list the p values you want as columns, pick the table type, and the grid builds as you type.

1. Number of trials, n. A whole number from 0 to 200. The table has n + 1 rows, one for each possible count.
2. Column probabilities, p. Separate values with commas or spaces. Decimals (0.25), fractions (1/6) and percentages (25%) are all accepted, and you can list up to 19 columns. The quick buttons fill in the classic textbook sets.
3. Table type. Exact P(X = x), cumulative P(X ≤ x), or upper-tail P(X ≥ x). Printed tables come in all three styles, so pick the one that matches the book you are checking against.
4. Look up one cell. Enter x and a single p. The result cards give the exact, cumulative and upper-tail probabilities for that cell, plus the mean and standard deviation, and the chart shades the outcomes up to x.

When you list only one p, the table switches to a full set of columns for that p: exact, at most, at least and fewer than. The copy button puts the whole grid on your clipboard, tab separated, ready to paste into Excel, Google Sheets or Word.

How Do You Read a Binomial Probability Table?

Find the table for your n, go down to the row for x, and across to the column for p. The number where they meet is the probability.

Take n = 10 and p = 0.30 and ask for exactly 3 successes. In the exact table, the row x = 3 and the column p = 0.30 meet at 0.2668. That is P(X = 3), about a 26.7% chance. The whole p = 0.30 column for n = 10 reads 0.0282, 0.1211, 0.2335, 0.2668, 0.2001, 0.1029, 0.0368, 0.0090, 0.0014, 0.0001 and 0.0000 for x = 0 to 10. The biggest value sits at x = 3, which is the mean np = 3, and the column adds up to 1 apart from rounding.

Before you read any value, check two things. First, is the table exact or cumulative? The same cell holds 0.2668 in one and 0.6496 in the other. Second, is it laid out by p across the top and x down the side, or the other way round? Older tables sometimes flip it. The title line, usually something like “Individual probabilities” or “Cumulative probabilities P(X ≤ x)”, settles the first question in a second.

How Do You Read a Cumulative Binomial Table?

A cumulative table gives P(X ≤ x), the chance of x successes or fewer, so the cell is the running total of the exact column down to that row.

For n = 10 and p = 0.30 the cumulative column reads 0.0282, 0.1493, 0.3828, 0.6496, 0.8497, 0.9527, 0.9894, 0.9984, 0.9999, 1.0000 and 1.0000. The last entry is always 1, because X is certain to be somewhere between 0 and n. If it is not 1.0000, you are probably looking at a table cut off at a small x, or one with a printing slip.

To get an exact probability from a cumulative table, subtract the cell above from the cell you want: P(X = x) = F(x) − F(x − 1). Here P(X = 4) = 0.8497 − 0.6496 = 0.2001, which matches the 0.2001 in the exact table. For x = 0 there is no cell above, so P(X = 0) is just F(0).

Reading across a row also tells you something. Moving right means a larger p, and larger p makes small counts less likely, so a cumulative row falls steadily as you go right. For n = 10 and x = 3 it drops from 0.9872 at p = 0.10 to 0.1719 at p = 0.50. If a row ever rises as you move right, you have misread the table.

How Do You Find At Least, More Than and Between From a Table?

Every wording can be turned into cell values with one rule: write the event as a difference of cumulative probabilities, then read the cumulative column. F(x) stands for the cumulative value P(X ≤ x).

With n = 10 and p = 0.50, “at least 7” is 1 − F(6) = 1 − 0.8281 = 0.1719. “Between 3 and 6 inclusive” is F(6) − F(2) = 0.8281 − 0.0547 = 0.7734. If you choose the upper-tail table type on this page, the “at least” column is printed for you, so no subtraction is needed.

Worked Example: Free Throws With n = 6 and p = 0.60

A player makes 60% of free throws and takes 6 independent shots. What is the probability of making fewer than 4?

The n = 6, p = 0.60 column of the exact table reads 0.0041, 0.0369, 0.1382, 0.2765, 0.3110, 0.1866 and 0.0467 for x = 0 to 6. Fewer than 4 means x = 0, 1, 2 or 3, so add the first four entries: 0.0041 + 0.0369 + 0.1382 + 0.2765 = 0.4557. A cumulative table would hand you the same number in one cell, F(3) = 0.4557.

The complement answers the opposite question. The chance of making 4 or more is 1 − 0.4557 = 0.5443, and the most likely single outcome is 4 makes at 0.3110. The mean is np = 3.6, so the mode of 4 is just above it, which is what you expect when p is above one half.

Worked Example: A Fair Coin, n = 20 and Exactly 12 Heads

Twenty fair coin flips give n = 20 and p = 0.50. How likely is exactly 12 heads, and how likely is 12 or more?

Many printed tables stop at n = 20 or 25, and this is one of the largest cases they cover. In the exact table the cell for x = 12 and p = 0.50 holds 0.1201, so P(X = 12) is about 12.0%. In the cumulative table, F(12) = 0.8684 and F(11) = 0.7483. The difference, 0.8684 − 0.7483, gives 0.1201 again, a useful self-check.

For “12 or more” use the complement of F(11): 1 − 0.7483 = 0.2517. About one run in four of 20 flips gives 12 or more heads. The mean is 10 and the standard deviation is about 2.24, so 12 is less than one standard deviation above the mean. That fits with a probability that is not small.

What if p Is Above 0.5? The Symmetry Rule

Many printed tables only show p up to 0.50 because the binomial distribution is mirror-symmetric in p: P(X = x) with success probability p equals P(X = n − x) with success probability 1 − p.

In words, counting successes at probability p is the same as counting failures at probability 1 − p. Take n = 10. P(X = 7) at p = 0.70 is 0.2668, and P(X = 3) at p = 0.30 is also 0.2668. The same trick works for cumulative values, with the direction flipped: P(X ≤ 4) at p = 0.70 is 0.0473, and P(X ≥ 6) at p = 0.30 is 0.0473.

To use a table that stops at 0.50 for p = 0.80, look up p = 0.20 and swap the roles of successes and failures. Read row n − x instead of row x, and for a cumulative question change “at most” to “at least”. This calculator skips the trick and builds the p = 0.80 column directly, but it is worth knowing when you are working from a paper table in an exam.

What Formula Fills Each Cell?

Each exact cell is the binomial probability mass function: choose which x of the n trials are successes, then multiply the probabilities of the successes and failures.

Here C(n, x) = n! / (x!(n − x)!) is the number of ways to place x successes among n trials. The same numbers are the entries of Pascal’s triangle, so the binomial expansion calculator will list C(n, x) for the whole row. Cumulative cells simply add the exact ones, F(x) = P(X = 0) + … + P(X = x), and the upper-tail cells add from x up to n.

For one cell, say n = 10, p = 0.30, x = 3: C(10, 3) = 120, 0.3³ = 0.027 and 0.7⁷ = 0.0823543, so 120 × 0.027 × 0.0823543 = 0.2668. When you build the table row by row, a shortcut helps: each entry follows from the one above by multiplying by (n − x)/(x + 1) × p/(1 − p). That is how this calculator fills the column, starting from the mode, and it is why very large n still works without overflow.

What Do the Mean, Variance and Mode Look Like in the Table?

The centre of the column is the mean np, the spread follows the variance np(1 − p), and the tallest entry sits at the mode, usually the integer part of (n + 1)p.

For n = 10 and p = 0.30 the mean is 3, the variance is 2.1 and the standard deviation is about 1.449. The column peaks at x = 3 with 0.2668, and the entries fall away on both sides, faster on the right because p is below one half, so the distribution leans toward small counts. For p = 0.50 the column is perfectly symmetric: 0.0010, 0.0098, 0.0439, 0.1172, 0.2051, 0.2461, then the same values in reverse order.

A quick sanity check for any table you build or copy: the exact column must sum to 1 (to rounding), the cumulative column must end at 1 and never decrease, and the largest exact entry should sit close to np. If one of those fails, the table or your reading of it has a problem. The binomial distribution calculator reports the mean, variance, standard deviation, mode and skewness directly if you want them without scanning the table.

How Accurate Are Rounded Table Values?

Printed tables round each cell to three or four decimals, and adding rounded cells can drift a little from the true total.

The n = 10, p = 0.30 exact column, rounded to four places, adds to 0.9999 rather than 1. Every individual cell is correct, but the rounding errors of up to 0.00005 per entry accumulate. That is normal, and it is why you should take a cumulative cell straight from a cumulative table when you can instead of summing exact cells yourself.

It also explains the occasional zero. A cell printed as 0.0000 is not exactly zero. For n = 10 and p = 0.30, P(X = 10) is 0.0000059, which rounds to nothing at four decimals. This page keeps the full-precision value behind each cell. Use the decimal places setting to show more digits, and the result cards for the cell you care about always show the value without table rounding, using scientific notation when it is tiny.

What Mistakes Do People Make With Binomial Tables?

• Mixing up exact and cumulative tables. Always read the heading. 0.2668 and 0.6496 are both legitimate entries for the same row and column.
• Off-by-one on “fewer than” and “at least”. “Fewer than 4” is F(3), not F(4). “At least 4” is 1 − F(3), not 1 − F(4).
• Using the wrong n. Each n has its own table. The row labelled 3 in the n = 8 table is not the row labelled 3 in the n = 10 table.
• Looking up p that is not on the grid. If p = 0.37 is not a column, interpolating between 0.35 and 0.40 is only approximate. Build the exact column here instead.
• Ignoring the independence check. A table is only valid if the trials are independent with a constant p. Drawing cards without replacement breaks that, and the hypergeometric calculator is the right model.
• Treating a rounded sum as a contradiction. A column that totals 0.9999 is fine.

What Do You Do When n Is Too Big for the Table?

Printed tables rarely go past n = 20 or 25. For larger n you can build the exact column here, or approximate with a normal or Poisson distribution when the conditions allow. Before you read any row, confirm the situation is binomial with the four binomial conditions: fixed n, two outcomes, constant p and independent trials.

The exact answer is always better when you have a tool. For comparison, take n = 50 and p = 0.10 and ask for P(X ≤ 3). The exact value is 0.2503. A normal approximation with a continuity correction, using the normal CDF calculator with mean 5 and standard deviation 2.121 at 3.5, gives 0.2398. Without the continuity correction it falls to 0.1729, which is why the correction matters. A Poisson approximation with λ = 5 gives 0.2650, because this is a case where p is small but n is not very large. Compare that with n = 100 and p = 0.50, where P(X ≤ 45) is 0.1841 exactly and 0.1841 by the corrected normal approximation, a much better fit since np and n(1 − p) are both large.

A common rule is that the normal approximation behaves well when np and n(1 − p) are both at least 10. The Poisson distribution is the usual choice when n is large and p is tiny. Neither is needed when the exact column takes a fraction of a second to build.

Binomial Table vs Calculator vs Software: Which Should You Use?

Use a printed table in an exam where it is allowed, a calculator or this page for homework and checking, and software when you are working with data.

In a spreadsheet you can rebuild any textbook table in a minute: put x in column A and p across row 1, then use BINOM.DIST with absolute references. The advantage of this page is that it also shows the upper tail, the mean and the chart, and it never makes you do the subtraction by hand.

Frequently Asked Questions

Sources: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.18 Binomial Distribution. SciPy documentation, scipy.stats.binom. R documentation, Binomial. Microsoft Support, BINOM.DIST function.

Related Calculators

These pages cover single probabilities, running totals and the whole distribution, or browse the full list of free statistics calculators:

Binomial CDF CalculatorRunning totals, at least, between and the inverse. Binomial Probability CalculatorExact, at most, at least and between probabilities. Binomial Distribution CalculatorMean, variance and the full distribution for any n and p. All StatsSolve CalculatorsBrowse the full directory of free statistics tools.