Using the Table of Binomial Probabilities

Statistics • Discrete Random Variables and Their Probability Distributions

Written by STEM Calculators Team Published December 13, 2025 Updated February 24, 2026

Using the Table of Binomial Probabilities

Many textbooks provide a binomial probability table (Table I) for n = 1 to 25 and selected values of p. This tool reproduces that idea: it shows the Table I slice for your n, lets you “read” P(x), and then uses those table entries to compute probabilities such as at most, at least, and between.

What to find

n (trials, 1 to 25)

Table I is intended for n ≤ 25.

p (success probability)

The table uses selected p values (commonly steps of 0.05).

x (successes)

Then n − x failures.

Rounding

Table entries are usually rounded; sums may differ slightly from calculator results.

Ready

Table I slice for your n (visualization)

Choose n and p, then Calculate to highlight the needed entries.

Bar graph of the probability distribution

Shows P(x) for x = 0, 1, 2, … , n (highlighted bars match your selected event).

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Using the binomial probability table

Many statistics textbooks include a table of binomial probabilities for n = 1 to 25 and selected values of p. Each table entry gives P(x), the probability of exactly x successes in n trials.

How to read a table entry

Identify n (number of trials).
Identify x (number of successes).
Choose the column for p (probability of success).
The entry at the intersection gives P(X = x) (usually rounded).

Common phrases and what they mean

Exactly x: P(X = x) (read one table cell).
At most x: P(X ≤ x) = P(0) + P(1) + … + P(x) (sum several cells).
At least x: P(X ≥ x) = P(x) + … + P(n) (sum several cells).
Between a and b: P(a ≤ X ≤ b) = P(a) + … + P(b).
Complement shortcut: P(X ≥ x) = 1 − P(X ≤ x − 1).

Where the table values come from

The table entries are based on the binomial model (independent trials; constant success probability). In formula form:

\[ P(X=x)=\binom{n}{x}\,p^{x}\,(1-p)^{\,n-x} \]

Tables typically round values (often to 4 decimals), so sums from the table can differ slightly from calculator output.

Frequently Asked Questions

How do I read a binomial probability from a textbook table?

Choose the row for n, the row for x (successes), and the column for p. The entry at the intersection gives P(X = x), usually rounded.

How do you use a binomial table to find P(X <= x) or P(X >= x)?

For P(X <= x), add the table entries from x = 0 up to x. For P(X >= x), add the entries from x up to n, or use the complement shortcut P(X >= x) = 1 - P(X <= x - 1).

Why can binomial table sums differ slightly from calculator results?

Table entries are typically rounded (often to 4 decimals), so adding multiple rounded values can introduce small rounding differences. Using the more detailed view reduces this effect.

What does P(a <= X <= b) mean in a binomial table problem?

It means the number of successes is between a and b inclusive. You compute it by summing the table entries P(X = a) + P(X = a + 1) + ... + P(X = b).

Using the Table of Binomial Probabilities

Reading the table + computation steps

Table I slice for your n (visualization)

Bar graph of the probability distribution

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Frequently Asked Questions

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