The binomial distribution formula calculates the probability of obtaining exactly $ x $ successes in $ n $ independent trials of a binomial experiment, where each trial has two possible outcomes (success or failure) and the probability of success $ p $ remains constant.

The formula is:

$P(X=x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x}$

Where:

• $ \binom{n}{x} = \frac{n!}{x!(n - x)!} $ is the binomial coefficient, representing the number of ways to choose $ x $ successes from $ n $ trials.

• $ p $ is the probability of success on a single trial.

• $ 1 - p $ (often denoted as $ q $) is the probability of failure.

• $ x $ is the number of successes (can be 0, 1, 2, ..., $ n $).

• $ n $ is the total number of trials.

This formula is used for discrete probability distributions and applies when:

• There are a fixed number of trials ($ n $).

• Each trial has only two outcomes: success or failure.

• Trials are independent.

• The probability of success ($ p $) is constant across trials.

Key Properties:

• Mean (Expected Value): $ \mu = np $

• Variance: $ \sigma^2 = np(1 - p) $

• Standard Deviation: $ \sigma = \sqrt{np(1 - p)} $

Example:

If a fair coin is tossed 10 times ($ n = 10 $), the probability of getting exactly 6 heads ($ x = 6 $, $ p = 0.5 $) is:

$P(X=6)=\left(\genfrac{}{}{0ex}{}{10}{6}\right)(0.5{)}^6(0.5{)}^4=210\times (0.5{)}^{10}\approx 0.205$

This formula is implemented in software like Excel using BINOM.DIST(x, n, p, FALSE) for exact probabilities and BINOM.DIST(x, n, p, TRUE) for cumulative probabilities.