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−k

When to use: Flip a biased coin n 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=0.375

Notation

X∼B(n,p) reads 'X follows a binomial distribution with n trials and success probability p'

What This Formula Means

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

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

Formal View

P(X=k)=(nk)pk(1−p)n−k for k=0,1,…,n; E(X)=np, Var(X)=np(1−p)

Worked Examples

Example 1

medium
A fair coin is flipped 8 times. What is the probability of getting exactly 5 heads?

Answer

P(X=5)=732≈0.219

First step

1
Use the binomial formula: P(X=k)=(nk)pk(1−p)n−k, where n=8, k=5, p=0.5.

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Example 2

hard
A multiple-choice quiz has 10 questions, each with 4 choices. If a student guesses randomly, what is the probability of getting at least 2 correct?

Example 3

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

Common Mistakes

  • Applying it when trials aren't independent or p changes - drawing without replacement is hypergeometric, not binomial.
  • Dropping the (nk) factor - you must count the orderings of the k successes, not just multiply pk(1−p)n−k.
  • Mismatching the exponents - the success power is k and the failure power is n−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 n independent yes/no trials, each with probability p.

How do you use the Binomial Distribution formula?

Flip a biased coin n 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) reads 'X follows a binomial distribution with n trials and success probability p'

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 p 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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