Binomial Distribution Formula

The Formula

P(X = k) = \binom{n}{k} p^k (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 \text{ heads}) = C(3, 2)(0.5)^2(0.5)^1 = \frac{3}{8} = 0.375

Notation

X \sim 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) = \binom{n}{k} p^k (1-p)^{n-k} for k = 0, 1, \ldots, n; E(X) = np, \text{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?

Solution

  1. 1
    Use the binomial formula: P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, where n = 8, k = 5, p = 0.5.
  2. 2
    \binom{8}{5} = \frac{8!}{5! \cdot 3!} = 56.
  3. 3
    P(X = 5) = 56 \times (0.5)^5 \times (0.5)^3 = 56 \times (0.5)^8 = 56 \times \frac{1}{256} = \frac{56}{256} = \frac{7}{32}.

Answer

P(X = 5) = \frac{7}{32} \approx 0.219
The binomial distribution models the number of successes in n independent trials, each with the same probability p of success. The combination \binom{n}{k} counts the number of ways to arrange k successes among n trials.

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?

Common Mistakes

  • Applying the binomial model when trials are not independent — drawing cards without replacement violates independence
  • Forgetting the \binom{n}{k} coefficient and computing only p^k(1-p)^{n-k}, which gives the probability of one specific sequence
  • Using the binomial distribution when the number of trials is not fixed — if counting trials until a success, use the geometric distribution

Why This Formula Matters

Models quality control, medical trials, polling, and any repeated yes/no experiment.

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 \sim 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?

Models quality control, medical trials, polling, and any repeated yes/no experiment.

What do students get wrong about Binomial Distribution?

The \binom{n}{k} counts the arrangements—without it you'd only get one specific order's probability.

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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