Binomial Distribution Formula

Binomial distribution is the probability distribution of the number of successes in n independent yes/no trials, each with probability p.

The Formula

P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}

When to use: Flip a biased coin nn times—how many heads? The binomial distribution gives the probability of each count.

Quick Example

Flip fair coin 3 times: P(2 heads)=C(3,2)(0.5)2(0.5)1=38=0.375P(2 \text{ heads}) = C(3, 2)(0.5)^2(0.5)^1 = \frac{3}{8} = 0.375

Notation

X∼B(n,p)X \sim B(n, p) reads 'XX follows a binomial distribution with nn trials and success probability pp'

What This Formula Means

The probability distribution of the number of successes in nn independent yes/no trials, each with probability pp.

Flip a biased coin nn times—how many heads? The binomial distribution gives the probability of each count.

Formal View

P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k} for k=0,1,…,nk = 0, 1, \ldots, n; E(X)=npE(X) = np, Var(X)=np(1−p)\text{Var}(X) = np(1-p)

Worked Examples

Example 1

medium
A fair coin is flipped 88 times. What is the probability of getting exactly 55 heads?

Answer

P(X=5)=732≈0.219P(X = 5) = \frac{7}{32} \approx 0.219

First step

1
Use the binomial formula: P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, where n=8n = 8, k=5k = 5, p=0.5p = 0.5.

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Example 2

hard
A multiple-choice quiz has 1010 questions, each with 44 choices. If a student guesses randomly, what is the probability of getting at least 22 correct?

Example 3

medium
A pharmacy fills 12 prescriptions; each has a 5% error rate independently. Find P(no errors)P(\text{no errors}).

Common Mistakes

  • Applying it when trials aren't independent or pp changes - drawing without replacement is hypergeometric, not binomial.
  • Dropping the (nk)\binom{n}{k} factor - you must count the orderings of the kk successes, not just multiply pk(1−p)n−kp^k(1-p)^{n-k}.
  • Mismatching the exponents - the success power is kk and the failure power is n−kn-k; swapping them inverts the answer.

Why This Formula Matters

The binomial is the workhorse model for 'repeated yes/no events' — defective parts, free-throw makes, survey yeses — and it's where the binomial coefficient earns its keep inside probability. Checking its four conditions teaches students to verify a model fits before plugging into a formula. Recognizing it by "Are there a fixed number of independent trials, each success/failure with the same probability, and I want a count of successes?" — rather than by familiar numbers — is what lets a student tell it apart from binomial coefficient and normal distribution and geometric / hypergeometric in a mixed problem set.

Frequently Asked Questions

What is the Binomial Distribution formula?

The probability distribution of the number of successes in nn independent yes/no trials, each with probability pp.

How do you use the Binomial Distribution formula?

Flip a biased coin nn times—how many heads? The binomial distribution gives the probability of each count.

What do the symbols mean in the Binomial Distribution formula?

X∼B(n,p)X \sim B(n, p) reads 'XX follows a binomial distribution with nn trials and success probability pp'

Why is the Binomial Distribution formula important in Math?

The binomial is the workhorse model for 'repeated yes/no events' — defective parts, free-throw makes, survey yeses — and it's where the binomial coefficient earns its keep inside probability. Checking its four conditions teaches students to verify a model fits before plugging into a formula. Recognizing it by "Are there a fixed number of independent trials, each success/failure with the same probability, and I want a count of successes?" — rather than by familiar numbers — is what lets a student tell it apart from binomial coefficient and normal distribution and geometric / hypergeometric in a mixed problem set.

What do students get wrong about Binomial Distribution?

The procedure for binomial distribution is the easy part; the trap is applying it when trials aren't independent or pp changes. Asking "Are there a fixed number of independent trials, each success/failure with the same probability, and I want a count of successes?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Binomial Distribution formula?

Before studying the Binomial Distribution formula, you should understand: binomial coefficient, probability, independent events.

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