How to find binomial probability (formula, examples)

What counts as a binomial probability

Binomial probability shows up whenever you repeat the same yes-or-no trial a fixed number of times and count how often the 'yes' outcome happens. Flipping a coin 10 times and counting heads is one example, and inspecting a batch of parts and counting defects is another, as long as the trials do not affect one another.

The count of successes is a random variable, usually called $X$. A binomial probability question asks for the chance that $X$ equals a particular value, or that it lands in a range such as 'at least 3' or 'at most 5'.

The binomial probability formula

A binomial probability is the chance of getting exactly $x$ successes in $n$ independent trials when each trial has the same success probability $p$. The formula on the AP Statistics formula sheet is:

Here $X$ is the random variable that counts successes, and $x$ is the specific count you are asking about. The letter $n$ is the number of trials, and $p$ is the probability of success on one trial, so $1-p$ is the probability of failure.

The piece $\binom{n}{x}$, read 'n choose x', counts how many different orders produce exactly $x$ successes, and you compute it as $\frac{n!}{x!(n-x)!}$. The factor $p^x$ is the chance of the $x$ successes, and $(1-p)^{n-x}$ is the chance of the remaining $n-x$ failures.

When a situation is binomial (the BINS checklist)

Before using the formula, check that the setup is really binomial. A quick checklist is BINS:

• Binary: each trial has only two outcomes, which you label success and failure.
• Independent: the result of one trial does not change the probability for any other trial.
• Number fixed: you set the number of trials $n$ before you start.
• Same probability: the success probability $p$ stays the same on every trial.

If any item fails, the formula does not apply. Dealing cards without replacement breaks independence, and waiting until your first success with no fixed $n$ is geometric rather than binomial. See binomial vs geometric for that comparison.

An exact binomial probability

To find $P(X = x)$ for a single exact count, put $n$, $p$, and $x$ into the formula and work out the three pieces separately before multiplying.

1. Find the binomial coefficient $\binom{n}{x}$.
2. Raise $p$ to the power $x$.
3. Raise $1-p$ to the power $n-x$.
4. Multiply the three results together.

Keeping the three factors separate makes the arithmetic easy to check and matches how a calculator breaks the work down. The first worked example below shows every step for a five-question guessing quiz.

'At least' and 'at most' probabilities (the complement trick)

Most questions ask for a range of counts rather than a single value. You can add the exact probabilities you need, or use a shortcut.

An 'at most' question such as $P(X \le 2)$ is a direct sum: $P(X = 0) + P(X = 1) + P(X = 2)$.

An 'at least' question such as $P(X \ge 1)$ is where the complement trick saves work. The opposite of 'at least 1' is 'exactly 0', so:

More generally, $P(X \ge k) = 1 - P(X \le k-1)$. Subtracting a few unwanted outcomes from 1 beats adding many terms by hand.

Translate the words into a boundary first: 'fewer than $k$' means $P(X \le k-1)$, and 'more than $k$' means $P(X \ge k+1)$. Getting the boundary right matters more than the arithmetic. The at least and at most worked examples below use both ideas.

Expected value and standard deviation of a binomial

The expected value of a binomial, written $\mu_X$ and read as the mean of $X$, is the long-run average number of successes. Its standard deviation, $\sigma_X$, measures the typical spread around that mean. Because the trials are identical and independent, both have short formulas:

For a player who takes 25 free throws at $p = 0.8$, the expected number of makes is $np = 25 \times 0.8 = 20$. The fourth worked example finishes the standard deviation. A larger $\sigma_X$ means the actual count is likely to land further from $np$ on any given run. The spread is small when $p$ is near 0 or 1, and it is widest at $p = 0.5$.

Binompdf and binomcdf on a calculator

A graphing calculator with statistics functions finds binomial probabilities directly, which is what you will use on the AP exam. Two functions do the work.

• A binomial pdf function, often written binompdf(n, p, x), returns the exact probability $P(X = x)$.
• A binomial cdf function, often written binomcdf(n, p, x), returns the cumulative probability $P(X \le x)$.

Match the function to the range in the question:

• Exactly $k$: binompdf(n, p, k).
• At most $k$: binomcdf(n, p, k).
• At least $k$: 1 - binomcdf(n, p, k-1).
• Fewer than $k$: binomcdf(n, p, k-1).

Enter $n$ first, then $p$, then the count. Our binomial probability calculator takes the same three inputs if you want to check your work without a handheld device.

Common mistakes to avoid

A few errors show up again and again:

• Reading the range wrong. 'At least 3' means $X \ge 3$ and includes 3, while 'more than 3' means $X > 3$ and starts at 4. Pin down the boundary before computing.
• Dropping the binomial coefficient. The product $p^x (1-p)^{n-x}$ by itself is the chance of one specific arrangement. Multiplying by $\binom{n}{x}$ counts every arrangement that gives $x$ successes.
• Forcing a non-binomial setup. If the trials are linked or the probability drifts, the formula returns a wrong answer. Sampling a small population without replacement, for instance, makes the outcome of one draw depend on the others.