Binomial Distribution Calculator (PMF)

What Is the Binomial Distribution?

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

Parameters:

• n = number of trials
• p = probability of success on each trial
• k = number of successes

The question: What's P(X = k), the probability of exactly k successes?

The Formula

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

Where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.

Example: Flip a fair coin 10 times. What's P(exactly 6 heads)?

$P(X = 6) = \binom{10}{6} \times 0.5^6 \times 0.5^4 = 210 \times 0.0156 \times 0.0625 = 0.205$

About 20.5% chance of exactly 6 heads.

When Does Binomial Apply?

The binomial distribution requires:

1. Fixed number of trials (n)
2. Two outcomes per trial: Success or failure
3. Constant probability (p): Same for each trial
4. Independence: Trials don't affect each other

Examples That Fit

• Coin flips (n flips, P(heads) = 0.5)
• Defective items in a sample (with replacement or large population)
• Survey responses (yes/no questions)
• Medical trials (treatment success/failure)
• Quality control (pass/fail tests)

Examples That Don't Fit

• Drawing without replacement from a small population (use hypergeometric)
• Number of events in a time interval (use Poisson)
• Varying success probabilities (use more complex models)

Applications

Quality Control

"A process has 5% defect rate. In a sample of 20, what's the probability of exactly 2 defectives?"

P(X = 2) = C(20,2) × 0.05² × 0.95¹⁸ = 190 × 0.0025 × 0.397 = 0.189

About 19% chance.

Medical Testing

"A drug has 80% efficacy. If 15 patients take it, what's P(exactly 12 respond)?"

P(X = 12) = C(15,12) × 0.8¹² × 0.2³ = 455 × 0.0687 × 0.008 = 0.250

Sports

"A basketball player makes 70% of free throws. In 8 attempts, what's P(exactly 6 made)?"

P(X = 6) = C(8,6) × 0.7⁶ × 0.3² = 28 × 0.118 × 0.09 = 0.296

Genetics

"Each child has 25% chance of inheriting a trait. For 4 children, what's P(exactly 1 has it)?"

P(X = 1) = C(4,1) × 0.25¹ × 0.75³ = 4 × 0.25 × 0.422 = 0.422

The Binomial Coefficient

$\binom{n}{k}$ counts the number of ways to choose k items from n:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Examples:

• C(5,2) = 10 (ways to choose 2 from 5)
• C(10,3) = 120
• C(n,0) = C(n,n) = 1
• C(n,1) = n

This is why it's called "binomial"—these coefficients appear in the binomial expansion of (a+b)ⁿ.

Distribution Shape

The shape depends on p:

• p = 0.5: Symmetric
• p < 0.5: Right-skewed (tail toward high k)
• p > 0.5: Left-skewed (tail toward low k)

As n increases, the binomial approaches normal:

• Mean = $np$
• Variance = $np(1-p)$
• SD = $\sqrt{np(1-p)}$

Rule of thumb: Normal approximation is reasonable when np ≥ 5 AND n(1-p) ≥ 5.

Expected Value and Variance

Mean (expected number of successes):

$E(X) = np$

Variance:

$Var(X) = np(1-p)$

Standard deviation:

$SD(X) = \sqrt{np(1-p)}$

For 100 coin flips (n=100, p=0.5):

• Mean = 50 heads
• SD = √(100 × 0.5 × 0.5) = 5 heads

Historical Development

The binomial distribution is one of the oldest probability distributions.

Blaise Pascal and Pierre de Fermat (1654) solved gambling problems that led to binomial thinking.

Jacob Bernoulli (1713) published the first comprehensive treatment in "Ars Conjectandi," including what we now call Bernoulli trials.

Abraham de Moivre (1733) discovered the normal approximation to the binomial—a key result leading to the Central Limit Theorem.

Cumulative Probability

For $P(X \leq k)$ or $P(X \geq k)$, use the Binomial Cumulative Calculator:

$P(X \leq k) = \sum_{i=0}^{k} P(X = i)$

This is often more useful than exact probabilities:

• "What's P(at most 3 defectives)?"
• "What's P(at least 8 successes)?"

Relationship to Other Distributions

Bernoulli: Binomial with n = 1

Normal: Binomial approaches normal for large n

Poisson: Binomial approaches Poisson when n is large and p is small (np = λ)

Negative Binomial: Number of trials until k successes

Hypergeometric: Binomial without replacement from finite population

Related Calculations

• Binomial Cumulative Calculator — P(X ≤ k), P(X ≥ k)
• Poisson Probability Calculator — For rare events
• Normal Probability Calculator — Continuous approximation
• Confidence Interval for Proportion — Estimate p

Summary

The binomial distribution gives the probability of exactly k successes in n independent trials, each with success probability p. From quality control to medical research to genetics, binomial probabilities answer fundamental questions about success counts. The formula combines the binomial coefficient (number of ways to arrange k successes) with the probability of any specific arrangement. For cumulative questions (at least k, at most k), use the cumulative calculator. For large n, the normal approximation provides a convenient alternative.