Binomial Distribution

Modeling the number of successes in a sequence of independent trials.

The "Series of Coin Flips" Distribution

The Binomial Distribution is a discrete probability distribution that models the number of successes in a fixed number of independent 'Bernoulli' trials, where each trial has only two possible outcomes: success or failure.

Think of flipping a coin 10 times and counting the number of heads. In finance, this could model the number of winning trades in a month (where each trade is a trial), or the number of portfolio companies that meet their earnings target in a quarter.

Interactive Binomial Distribution

Adjust the number of trials (n) and the probability of success (p) to see how the shape of the distribution changes.

Number of Trials (n): 20

Probability of Success (p): 0.50

Mean (

\mu

): 10.00

Variance (

\sigma^2

): 5.00

Core Concepts

Probability Mass Function (PMF)

The PMF answers: 'What is the probability of getting exactly k successes in n trials?'

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

To find the probability of exactly $k$ successes, we consider three pieces:

$p^k$ : This is the probability of achieving $k$ successes. If you want 3 wins and the probability of a win is $p$ , the combined probability is $p \times p \times p = p^3$ .
$(1-p)^{n-k}$ : This is the probability of getting the remaining $n-k$ outcomes as failures. The probability of one failure is $1-p$ .
$\binom{n}{k}$ : This is the binomial coefficient, read as "n choose k". It counts the number of different ways to arrange $k$ successes among $n$ trials. For example, in 4 trials, getting 2 successes (SSFF) could happen as SSFF, SFSF, SFFS, FSSF, FSFS, or FFSS. There are $\binom{4}{2}=6$ ways.

Hover over the formula components to see how they relate to the chart above.

Mean (

\mu

): 10.00

Variance (

\sigma^2

): 5.00

Cumulative Distribution Function (CDF)

The CDF answers: 'What is the probability of getting k successes or fewer?'

F(k) = P(X \le k) = \sum_{i=0}^{k} \binom{n}{i} p^i (1-p)^{n-i}

The CDF is simply the sum of all the probabilities from the PMF for all outcomes up to and including `k`. For example, the probability of getting 2 or fewer successes, $P(X \le 2)$ , is calculated as $P(X=0) + P(X=1) + P(X=2)$ .

Mean (

\mu

): 10.00

Variance (

\sigma^2

): 5.00

Expected Value & Variance

Expected Value (Mean)

E[X] = np

This is intuitive: if you flip a fair coin (p=0.5) 20 times (n=20), you would expect to get 20 * 0.5 = 10 heads on average.

Variance

Var(X) = np(1-p)

The variance measures the spread of the outcomes. Notice it's maximized when p=0.5 (a 50/50 coin flip has the highest uncertainty) and is 0 when the outcome is certain (p=0 or p=1).

Key Derivations

Deriving the Mean and Variance

The easiest way to derive the moments of a Binomial distribution is to recognize it as a sum of `n` independent Bernoulli trials.

Deriving the Expected Value (Mean)

Step 1: Decompose into Bernoulli Variables

A Binomial random variable $X$ with parameters $n$ and $p$ is the sum of $n$ independent Bernoulli random variables $Y_i$ , each with probability $p$ .

X = Y_1 + Y_2 + \dots + Y_n

We know from the Bernoulli distribution page that the mean of a single trial is $E[Y_i] = p$ .

Step 2: Use Linearity of Expectation

The expectation of a sum is the sum of the expectations. This property holds even if the variables are not independent.

E[X] = E[Y_1 + Y_2 + \dots + Y_n] = E[Y_1] + E[Y_2] + \dots + E[Y_n]

Step 3: Sum the Bernoulli Means

We are adding the same mean, $p$ , to itself $n$ times.

E[X] = \sum_{i=1}^{n} p = np

Final Mean Formula

E[X] = np

Deriving the Variance

We use the same decomposition. Since the Bernoulli trials are **independent**, the variance of their sum is the sum of their variances.

Step 1: Sum the Variances of Bernoulli Variables

The variance of a single Bernoulli trial is $Var(Y_i) = p(1-p)$ .

Var(X) = Var(Y_1 + \dots + Y_n) = Var(Y_1) + \dots + Var(Y_n)

Step 2: Final Result

We are adding the same variance to itself $n$ times.

Var(X) = \sum_{i=1}^{n} p(1-p) = np(1-p)

Final Variance Formula

Var(X) = np(1-p)

Applications

Quantitative Finance: Option Pricing

The Binomial Option Pricing Model uses a discrete-time binomial tree to model the price of an underlying asset. At each step, the price can move up or down with a certain probability `p`. By working backward from the option's expiration date, traders can calculate the option's fair value today. Each step in the tree is a Bernoulli trial, and the final distribution of possible prices follows a Binomial distribution.