Binomial Approximation - Mathematics.lu

This page can be left as background knowledge and a perhaps an additional motivation for studying both binomial variables (from S6) and normal variables.

A binomial random variable $X \sim B(n, p)$ counts the number of successes in $n$ independent trials. We can think of each individual trial as a mini-random variable (a Bernoulli variable). A binomial variable is simply the sum of many independent Bernoulli random variables, so the Central Limit Theorem (not part of your programme, but it was mentioned in a previous page) dictates that as the number of trials $n$ grows large, its discrete distribution naturally smooths out and mimics the shape of a continuous normal curve.

Table of Contents

Motivation

Calculating a precise binomial probability requires using the binomial formula:

$\mathbb{P}(X=k)={n \choose k}p^{k}(1-p)^{n-k}=\frac{n!}{k!(n-k)!}p^{k}(1-p)^{n-k}$

Now imagine you are a quality control manager at a factory, or a pharmaceutical scientist running a clinical trial, and you need to find the probability that at least 600 out of 1,000 patients recover, so $\mathbb{P}(X \ge 600)$ .

The Manual Nightmare: To do this exactly, you would have to calculate the probability for X=600, X=601, X=602, all the way up to X=1000, and add them together. That is 401 separate calculations.
The Factorial Explosion: Calculating factorials like 1000! is practically impossible by hand and causes standard software or older scientific calculators to crash due to “overflow errors” (the numbers simply become too massive for a system’s memory to process).

By swapping the binomial distribution for a normal curve, you replace hundreds of explosive factorial additions with a single integral.

While modern calculators can handle simple cumulative lookups instantly, normal approximation remains foundational to engineering, programming, and mathematical theory for several critical reasons. Therefore, binomial approximation is still a relevant tool to simplify calculation demands.

When Can we Approximate?

You cannot always swap one for the other. The approximation is only accurate when the binomial distribution becomes highly symmetric and has a negligible chance of being “cut off” by its natural boundaries $0$ and $n$ ).

To use the normal approximation safely, you must check the Success-Failure Condition:

$np \ge 5$ (The expected number of successes must be at least 5)
$n(1-p) \ge 5$ (The expected number of failures must be at least 5)

Some sources require these values to be at least 10, but 5 is the standard threshold for introductory statistics.

How to Do the Approximation

If the conditions above are met, you can map the binomial parameters directly to a normal distribution layout.

Match the Parameters: Calculate the mean and standard deviation from the binomial variables to use as your normal parameters:
$\mu =n\cdot p\text{ and }\sigma =\sqrt{n\cdot p\cdot(1-p)}$

You then define your approximating normal variable as: $Y \sim N(\mu, \sigma^2)$
Apply the Continuity Correction: Because you are using a continuous curve to estimate a discrete blocky bars, you must adjust the boundaries by 0.5 to capture the full width of the discrete values.
- To find the binomial probability $\mathbb{P}(X \le 10)$ , we evaluate the continuous normal probability $\mathbb{P}(Y \le 10.5)$ .
- To find the binomial probability $\mathbb{P}(X \ge 15)$ , we evaluate the continuous normal probability $\mathbb{P}(Y \ge 14.5)$ .

Example: A random sample of $n = 400$ adults is selected from a city where $65\%$ of the population are vaccinated. Let $X$ be the number of vaccinated individuals in the sample, where $X \sim B(400, 0.65)$ . Calculate $\mathbb{P}(X \ge 275)$ using both the exact binomial distribution and the continuous normal approximation to compare their accuracy.

Exercises

A high-tech manufacturing plant monitors screen defects on smartwatches. A batch contains a random sample of $n = 50$ screens, where the probability of a defect is $p = 0.04$ , meaning $X \sim B(50, 0.04)$ . Solve the following sub-problems:

1. Test the success-failure condition to determine if a normal approximation is safely applicable.
2. Calculate the exact binomial probability of finding at most 3 defective screens: $\mathbb{P}(X \le 3)$ .
3. Regardless of your answer to sub-problem 1, compute the estimated probability $\mathbb{P}(X \le 3)$ using a continuous normal approximation with a continuity correction.
4. Compare your results and evaluate why the approximation behaves this way in relation to the success-failure threshold.

An agricultural laboratory tests a strain of organic heirloom seeds. In a random sample of $n = 150$ planted seeds, the probability of successful germination is $p = 0.80$ , meaning $X \sim B(150, 0.80)$ . Solve the following sub-problems:

1. Verify if the success-failure condition is met to support a normal approximation.
2. Compute the exact binomial probability that fewer than 115 seeds germinate: $\mathbb{P}(X \le 114)$ .
3. Use a continuous normal approximation with a continuity correction to estimate the probability $\mathbb{P}(X \le 114)$ .
4. Compare both numbers and comment on the accuracy of the normal approximation.

An e-commerce platform monitors payment gateway failures. Out of a random sample of $n = 200$ customer checkouts, the probability of a transaction succeeding smoothly is $p = 0.98$ , meaning $X \sim B(200, 0.98)$ . Solve the following sub-problems:

1. Evaluate whether a normal approximation is appropriate using the success-failure condition rules.
2. Calculate the exact discrete binomial probability that at least 194 checkouts succeed: $\mathbb{P}(X \ge 194)$ .
3. Regardless of your answer to sub-problem 1, estimate the probability $\mathbb{P}(X \ge 194)$ using the continuous normal curve framework with a continuity correction.
4. Contrast your findings and comment on how a heavily skewed success rate near 1 impacts the absolute error of your approximation.

An urban charging grid maps out driver arrival habits. Out of a random selection of $n = 300$ electric vehicle drivers, the probability that a driver plugs into a fast-charger port is $p = 0.45$ , meaning $X \sim B(300, 0.45)$ . Solve the following sub-problems:

1. Use the standard thresholds to establish whether a normal approximation is valid for this size.
2. Calculate the exact binomial probability that at least 145 drivers choose a fast-charger port: $\mathbb{P}(X \ge 145)$ .
3. Use a continuous normal approximation with a continuity correction to find the estimated probability $\mathbb{P}(X \ge 145)$ .
4. Compare both final probability metrics and document the performance accuracy of the continuous curve.