Binomial Probability Formula

📊 intermediate

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

The probability of exactly $k$ successes in $n$ independent Bernoulli trials.

Variables & Units

$n$

Total number of trials

Unit: Dimensionless

$p$

Probability of success in a trial

Unit: Ratio

$k$

Number of successes

Unit: Dimensionless

Key Information

📐 Derived From

Combinatorial selection and the multiplication rule of probability.

💡 Example

Probability of getting 3 heads in 5 coin tosses.

🔧 Applications

Quality control, genetics, and clinical trials.

⚠️ Constraints

Trials must be independent and $p$ must be constant.

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