Binomial Distribution Formula

The binomial distribution formula calculates the probability of obtaining exactly $ x $ successes in $ n $ independent Bernoulli trials, where each trial has a success probability $ p $ and failure probability $ q = 1 - p $. 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 $ is the probability of failure,

• $ x $ is the number of successes (an integer from 0 to $ n $).

Key Properties

• Mean (Expected Value): $ \mu = np $

• Variance: $ \sigma^2 = np(1 - p) $

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

Applications

• Modeling outcomes in experiments with two possible results (e.g., heads/tails, pass/fail).

• Quality control, survey responses, and medical trials.

Example

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

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

In Excel

Use the BINOM.DIST function:

• BINOM.DIST(x, n, p, FALSE) for exactly $ x $ successes,

• BINOM.DIST(x, n, p, TRUE) for at most $ x $ successes.