Poisson Approximation to the Binomial Distribution

Definition

If a random variable $X$ follows a binomial distribution with parameters $n$ and $p$, written as

$$X \sim \text{Bin}(n,p),$$

then for large $n$ and small $p$, with the product

$$\lambda = np$$

remaining moderate, the binomial distribution can be approximated by a Poisson distribution with mean $\lambda$:

$$X \approx \text{Poisson}(\lambda).$$

Thus,

$$P(X=k) \approx \frac{e^{-\lambda}\lambda^k}{k!}, \quad k=0,1,2,\dots$$

This is called the Poisson approximation to the binomial distribution.

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Main Content

1. First Concept: Binomial Distribution and Its Exact Probability

• The binomial distribution models the number of successes in $n$ independent trials, where each trial has only two outcomes: success or failure.
• Its exact probability mass function is

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

where:

• $n$ = number of trials
• $p$ = probability of success in each trial
• $k$ = number of successes

The binomial model is exact, but for very large $n$, evaluating $\binom{n}{k}$ and the powers can be cumbersome.

Example:
If a factory produces thousands of items and the probability that an item is defective is very small, the number of defectives among many items follows a binomial distribution.

Why this matters:
The Poisson approximation begins with this binomial model and replaces it with a simpler model when conditions are suitable.

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2. Second Concept: Conditions for Poisson Approximation

• The approximation is valid when:
• $n$ is large,
• $p$ is small,
• $np=\lambda$ is moderate (not too large).

This means the event of success is rare, but there are many opportunities for it to occur.

A common rule of thumb is that the Poisson approximation is reasonable when:

• $n \ge 20$ and $p \le 0.05$, or
• $n$ is large enough that $np$ stays in a manageable range.

Interpretation:
When each individual trial has a very low probability of success, the total number of successes behaves like a count of rare events. The Poisson distribution is designed precisely for such counts.

Simple visual idea:

Trials: - - - - - - - - - - - - - - - -

Rare successes: - - S - - - - - S - - - - - -

Here, successes are sparse and scattered, which is the kind of pattern Poisson approximates well.

Example:
If the probability of a machine failure on a given day is 0.002 and you observe 1000 days, then $np = 1000 \times 0.002 = 2$. Since the event is rare and the number of trials is large, Poisson approximation can be used.

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3. Third Concept: Derivation of the Poisson Approximation

• Starting from the binomial probability formula,

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

we let

$$p=\frac{\lambda}{n}.$$

Then,

$$P(X=k)=\binom{n}{k}\left(\frac{\lambda}{n}\right)^k\left(1-\frac{\lambda}{n}\right)^{n-k}.$$

As $n$ becomes very large:

• $\left(1-\frac{\lambda}{n}\right)^n \to e^{-\lambda}$
• the combinatorial term behaves like $\frac{n^k}{k!}$ for fixed $k$

Combining these results gives:

$$P(X=k)\approx \frac{e^{-\lambda}\lambda^k}{k!}.$$

This is exactly the probability mass function of the Poisson distribution with mean $\lambda$.

Meaning of the derivation:
The approximation works because many tiny probabilities across many trials accumulate into a distribution governed by the average rate $\lambda$, not by the exact value of $n$ and $p$ separately.

Key insight:
The binomial distribution counts successes from repeated independent Bernoulli trials, while the Poisson distribution counts rare events over a fixed interval or number of opportunities.

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Working / Process

1. Identify whether the binomial setting is appropriate

• Check if the problem involves a fixed number of independent trials.
• Confirm that each trial has two outcomes: success or failure.
• Determine whether the success probability $p$ is small and $n$ is large.

2. Compute the Poisson parameter

• Find $\lambda = np$.
• This value becomes the mean of the approximating Poisson distribution.
• Use $\lambda$ in the Poisson formula instead of the full binomial expression.

3. Use the Poisson probability formula

• For exactly $k$ successes,

$$P(X=k) \approx \frac{e^{-\lambda}\lambda^k}{k!}.$$

• For probabilities like “at most 1” or “at least 2,” add the relevant Poisson terms.
• Compare the result with the exact binomial probability if needed to judge accuracy.

Worked example:
Suppose $X \sim \text{Bin}(200, 0.01)$. Find $P(X=3)$.

• Here, $n=200$, $p=0.01$
• $\lambda = np = 200 \times 0.01 = 2$
• Approximate using Poisson:

$$P(X=3)\approx \frac{e^{-2}2^3}{3!} = \frac{e^{-2}\cdot 8}{6} = \frac{4}{3}e^{-2} \approx 0.1804$$

This is much easier than computing the exact binomial expression.

For cumulative probability:
If you need $P(X\le 1)$,

$$P(X\le 1)\approx P(X=0)+P(X=1) = e^{-2}+\frac{2e^{-2}}{1!} = 3e^{-2} \approx 0.4060$$

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Advantages / Applications

Simplifies calculations

• The Poisson formula is much easier to use than the binomial formula when $n$ is very large.
• It avoids complicated binomial coefficients and large power computations.

Useful for rare events

• It is ideal for modeling events with small success probabilities.
• Common examples include defects, breakdowns, accidents, and misprints.

Widely used in real-life problems

• Quality control: number of defective products in a large batch
• Insurance: number of claims in a period
• Telecommunications: number of calls arriving in a time interval
• Biology/medicine: number of rare mutations or infections
• Traffic/safety: number of accidents in a region or time period

Example applications:

• If a newspaper has a very small chance of a printing error on each page and thousands of pages are printed, the number of errors can be approximated by Poisson.
• If a website receives a huge number of visits and the chance of a server error on each request is tiny, the total errors may also be approximated this way.

Why it is important in practice:
The approximation allows analysts to make fast, reliable estimates without heavy computation, especially in large-scale systems.

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Summary

• The Poisson approximation is used when a binomial distribution has large $n$ and small $p$.
• It replaces $\text{Bin}(n,p)$ with $\text{Poisson}(\lambda)$, where $\lambda = np$.
• It is especially effective for modeling rare events and simplifies probability calculations.

Important terms to remember:
Binomial distribution, Poisson distribution, rare events, approximation, probability mass function, $\lambda=np$, large $n$, small $p$