A binomial distribution table is a shortcut that gives you pre-calculated probabilities so you don’t have to plug numbers into the binomial formula by hand. To use one, you need three pieces of information: the number of trials (n), the probability of success on each trial (p), and the number of successes you’re interested in (k or x). You locate n and p in the table, find the row for your number of successes, and read off the probability.
How the Table Is Organized
Most printed binomial tables are grouped first by the number of trials, n. Within each n block, the rows represent the number of successes (0, 1, 2, … up to n), and the columns represent different values of p, the probability of success on a single trial. Standard tables typically cover n values from 1 through 20 or 25, and p values from 0.05 to 0.50, increasing in steps of 0.05.
To look up a probability, find the section for your n, move down to the row matching your desired number of successes, then slide across to the column for your p value. The number at that intersection is your answer.
Individual vs. Cumulative Tables
There are two types of binomial tables, and mixing them up is the most common mistake people make. An individual probability table gives you P(X = k), the probability of getting exactly k successes. A cumulative probability table gives you P(X ≤ k), the probability of getting k or fewer successes. You can usually tell which type you have by checking the label or notation: a lowercase f(x) signals individual probabilities, while an uppercase F(x) signals cumulative ones.
Cumulative tables are more versatile because you can derive individual probabilities from them. To find the probability of exactly 3 successes, subtract the cumulative probability at 2 from the cumulative probability at 3:
P(X = 3) = P(X ≤ 3) − P(X ≤ 2)
If your textbook or course provides a cumulative table, this subtraction trick is essential.
Finding “At Least,” “At Most,” and “Between” Probabilities
Most real problems don’t ask for the probability of exactly k successes. They ask things like “what’s the probability of getting at least 3?” or “at most 5?” Here’s how to handle each type.
At most k successes, P(X ≤ k): If you have a cumulative table, read the value directly. If you have an individual table, add up P(X = 0) + P(X = 1) + … + P(X = k).
At least k successes, P(X ≥ k): Use the complement. Since all probabilities sum to 1, P(X ≥ k) = 1 − P(X ≤ k − 1). For example, with 5 coin flips and p = 0.50, the probability of getting 2 or more heads is P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) = 0.3125 + 0.3125 + 0.1562 + 0.0313 = 0.8125. With a cumulative table, you’d simply calculate 1 − P(X ≤ 1).
Between two values, P(a ≤ X ≤ b): With a cumulative table, subtract: P(X ≤ b) − P(X ≤ a − 1). With an individual table, add up each probability from a to b.
What to Do When p Is Greater Than 0.50
Most tables only list p values up to 0.50. If your problem has a higher probability of success, you need to flip the problem using the symmetry of the binomial distribution. The core idea: getting k successes with probability p is the same as getting n − k failures with probability 1 − p.
Here’s the rule: substitute 1 − p for p, and substitute n − k for k. For example, if you need the probability of exactly 1 success in 4 trials with p = 0.60, look up n = 4, p = 0.40, and k = 3 instead. You’ll get the same answer.
This works for range problems too. Say you need P(X ≥ 3) with n = 7 and p = 0.85. Switch to p = 1 − 0.85 = 0.15, and convert the inequality: P(X ≥ 3) becomes P(X ≤ 7 − 3) = P(X ≤ 4) with the new p of 0.15. For “at most” problems, the conversion goes in the other direction: P(X ≤ 8) with n = 9 and p = 0.60 becomes P(X ≥ 9 − 8) = P(X ≥ 1) with p = 0.40.
A Worked Example
About 8% of males are colorblind. A researcher needs colorblind participants and checks 9 men. What’s the probability she finds 3 or more who are colorblind?
This is a binomial problem with n = 9, p = 0.08, and you need P(X ≥ 3). Since adding up P(3) + P(4) + … + P(9) requires seven lookups, use the complement instead: P(X ≥ 3) = 1 − P(X ≤ 2) = 1 − [P(0) + P(1) + P(2)].
Go to the table section for n = 9. Find the column for p = 0.08 (or the closest value your table provides). Read the probabilities for k = 0, k = 1, and k = 2, add them, and subtract from 1. The result is approximately 0.0298, or about a 3% chance. That low probability tells the researcher it would be fairly unusual to find three or more colorblind men in such a small sample.
Another Example: Coin Flips
Is it unusual to see fewer than 3 heads in 12 flips of a fair coin? Here n = 12, p = 0.50, and you need P(X ≤ 2) = P(0) + P(1) + P(2). Looking these up in the table for n = 12 and p = 0.50, the sum comes to roughly 0.019, or about 2%. Getting 2 or fewer heads in 12 flips happens less than 2% of the time, so yes, it would be genuinely unusual.
Limitations of Printed Tables
Standard binomial tables cap out at n = 20 or n = 25, depending on the textbook. If your problem involves more trials than that, the table won’t help. For larger n, you can use a calculator, statistical software, or the normal approximation to the binomial distribution (which works well when both np and n(1 − p) are at least 5).
Tables also only include p values at fixed intervals, typically every 0.05. If your probability of success falls between listed values (say p = 0.37), you won’t find an exact match. In that case, you’ll need the binomial formula, a graphing calculator, or software. Most statistics courses that rely on tables design their problems to land on the p values the table provides.
How the Table Relates to the Formula
Every value in a binomial table comes from the same formula you’d use by hand. That formula multiplies three things together: the number of ways to arrange k successes among n trials, the probability of success raised to the kth power, and the probability of failure raised to the (n − k)th power. The table simply saves you from computing this for every combination of n, k, and p. When you look up a value, you’re reading the result of that calculation, pre-done and rounded to four decimal places.
Understanding this connection helps when you need to verify a table reading or handle a problem the table can’t cover. The table is a convenience tool, not a different method. It gives you the same answer the formula would, just faster.