Binomial Distribution

ABSTRACT

The Binomial Distribution models the number of successes in a fixed number of independent Bernoulli Trials, each with the same probability of success $p$. Unlike a uniform distribution, the outcomes are not necessarily equally likely.

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Bernoulli Trial

A Bernoulli Trial is a performance of an experiment with exactly two possible outcomes (e.g., flipping a coin, a part being defective or non-defective).

• Success with probability $p$.
• Failure with probability $1-p$.

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Binomial Distribution Formula

For a particular number of trials $n$ and probability $p$, the sample space is the set of integers $\{0,1,2,\dots ,n\}$. The probability of achieving exactly $k$ successes is:

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

Understanding the Components

• $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$: The number of ways to choose which $k$ trials out of $n$ result in success.
• $p^k$: The probability that $k$ specific trials result in success.
• $(1-p{)}^{n-k}$: The probability that the remaining $n-k$ trials result in failure.

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Examples

1. Fair Coins (Uniform Case)

When flipping $n$ fair coins, the probability of getting exactly $k$ Heads ($H$) is:

$$P(k\text{ Hs})=\frac{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}{2^n}$$

NOTE

Here, $p=0.5$ and $(1-p)=0.5$. Since $0.5^k\cdot 0.5^{n-k}=0.5^n$, which is $1/2^n$, the formula simplifies to the ratio of successful sequences over total possible sequences ($2^n$).

2. Biased Trials (Non-Uniform Case)

What if the coin isn’t fair? If a biased coin has $P(H)=0.6$ and you flip it 10 times, the probability of getting exactly 7 Heads is:

$$P(7)=\left(\genfrac{}{}{0ex}{}{10}{7}\right)(0.6{)}^7(0.4{)}^3$$

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Properties and Analysis

Property	Formula
Sample Space	$S=\{0,1,2,\dots ,n\}$
Expected Value	$E[X]=np$
Variance	$Var(X)=np(1-p)$

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Related Notes

• Independent Events: Trials must be independent for this distribution to apply.
• Expected Value: The average number of successes over $n$ trials.
• Linearity of Expectation: Used to easily derive $E[X]=np$ by summing $n$ indicator variables.
• Geometric Distribution: Contrast this with the distribution where the number of trials is not fixed, but continues until the first success.