Binomial Distribution

TL;DR

Gives the probability of observing exactly x successes in n independent trials when each trial has the same success probability p.
Computed using the binomial coefficient multiplied by p^x and (1 − p)^(n − x).
Curve shape depends on n and p: symmetric for p = 0.5 (e.g., fair coin), skewed otherwise.

Definition

The binomial distribution is a probability distribution that describes the likelihood of a specific number of successes in a given number of independent trials. It uses the binomial coefficient to count the number of ways x successes can occur in n trials.

The probability of exactly x successes in n trials is given by:

P(x) = \frac{n!}{x!(n-x)!}\, p^x (1-p)^{\,n-x}

Where:

P(x) is the probability of x successes in n trials
n is the total number of trials
x is the number of successes
p is the probability of success in each trial
1 − p is the probability of failure in each trial

Explanation

The binomial coefficient, (\frac{n!}{x!(n-x)!}), counts how many distinct sequences of n trials contain exactly x successes.
Each distinct sequence with x successes and n − x failures has probability (p^x(1-p)^{n-x}) because trials are independent and each success (failure) contributes a factor p ((1 − p)).
Multiplying the count of sequences by the probability of any one such sequence yields P(x).
The binomial distribution is typically shown as a discrete probability mass function (a “binomial distribution curve”). For p = 0.5 and moderate n, the curve is symmetric with a central peak near the mean; for p ≠ 0.5 it is skewed (e.g., skewed to the right when p < 0.5).

Examples

Coin flips (n = 10, p = 0.5)

For flipping a fair coin 10 times, p = 0.5 and the formula becomes:

P(x) = \frac{10!}{x!(10-x)!}\, 0.5^x \, 0.5^{10-x}

Using this formula, the probability of each possible number of heads is:

P(0) = 0.000976562
P(1) = 0.009765625
P(2) = 0.043945312
P(3) = 0.117187500
P(4) = 0.205078125
P(5) = 0.24609375
P(6) = 0.205078125
P(7) = 0.117187500
P(8) = 0.043945312
P(9) = 0.009765625
P(10) = 0.000976562

The binomial distribution curve for this scenario is symmetrical, with the peak in the middle and probabilities decreasing as the number of heads deviates from the mean.

Sales calls (n = 100, p = 0.25)

Suppose a salesperson makes 100 sales calls and the probability of a customer making a purchase on any call is 0.25. The binomial distribution describes the likelihood of making 0, 1, 2, …, 100 sales. Using the binomial formula, example probabilities listed for this scenario are:

P(0) = 0.00000000000000028819816
P(1) = 0.00000000000028819816
P(2) = 0.00000000005764355328
P(3) = 0.000000005764355328
P(4) = 0.00000028819816
P(5) = 0.00000843454912
P(6) = 0.00018647163
P(7) = 0.00294912064
P(8) = 0.02893853
P(9) = 0.17684625
P(10) = 0.12093235
P(11) = 0.35486064
P(12) = 0.57643553
P(13) = 0.65346901
P(14) = 0.57643553
P(15) = 0.35486064
P(16) = 0.12093235
P(17) = 0.02893853
P(18) = 0.00294912064
P(19) = 0.00018647163
P(20) = 0.00000843454912
P(21) = 0.00000028819816
P(22) = 0.000000005764355328
P(23) = 0.00000000005764355328
P(24) = 0.00000000000028819816
P(25) = 0.00000000000000028819816

The binomial distribution curve for this scenario is described as skewed to the right, indicating a higher probability of making a smaller number of sales and a lower probability of making a larger number of sales.

Notes or pitfalls

The shape of the binomial distribution depends on both n and p.
For p = 0.5 the distribution is symmetric (peak near the mean); for p ≠ 0.5 it becomes skewed.
The distribution assumes independent trials and a constant probability of success p across trials.

Binomial coefficient
Binomial distribution curve
Mean