How the Central Limit Theorem Solves Real-World Probability Problems

Central Limit Theorem

Let X₁, X₂, … be independent and identically distributed random variables with finite mean μ and variance σ²; then for any real number z, P\left(\frac{\sum_{i=1}^{n}X_i - nμ}{\sigma\sqrt{n}} \le z\right) \approx \Phi(z), where \Phi is the standard normal distribution function.

This theorem states that if X₁,…,Xₙ are i.i.d., then when n is sufficiently large (generally at least 30, the larger the better), the sum (or average) of the variables approximates a standard normal distribution, or more generally a normal distribution.

Example 1

In a workshop there are 200 identical lathes, each requiring 10 kW. Each lathe is independently on with probability p. How much power must be supplied so that the probability that all lathes can operate is at least a given level?

Solution: Let Xᵢ denote the power usage of the i‑th lathe; the total power S = ΣXᵢ. By the CLT, S is approximately normal, leading to a required power of … kW. Supplying 2000 kW guarantees operation; to achieve the target probability only … kW is needed, saving … kW (about …% of total power).

Example 2

A city of one million people has a probability of 1/20 000 that a person needs an ambulance. How many ambulances must the emergency center have to ensure a probability of at least a given level that a call can be answered in time?

Solution: Let Yᵢ be the indicator that the i‑th person needs an ambulance. By the CLT the sum ΣYᵢ is approximately normal, giving a required number of ambulances equal to 67.

Example 3

A supermarket runs a prize draw: for every 50 ¥ spent a customer receives one ticket. Among 10 000 tickets there are 1 first prize, 10 second prizes, 100 third prizes, and 1 000 encouragement prizes (prizes can be combined). (1) If a person spends 3000 ¥, what is the probability of winning at least three prizes? (2) How much must a customer spend to have at least a given probability of winning a second‑prize or better?

Solution: By modeling the number of winning tickets as a sum of independent Bernoulli variables and applying the CLT, the required spending amount is found to be a very large number.

Example 4

When a computer adds numbers after rounding each to the nearest integer, the rounding errors are uniformly distributed on (‑0.5, 0.5). Assuming independence, for 1500 numbers what is the probability that the absolute total error is less than 15?

Solution: Let Eᵢ be the rounding error of the i‑th number. The sum S = ΣEᵢ has variance 1500·(1/12). By the CLT, S is approximately normal, yielding P(|S|<15) ≈ 0.125.

Example 5

In a non‑related donor pool, the probability of a bone‑marrow match is 1/100 000. For a leukemia patient: (1) With a registry of 200 000 donors, what is the probability of finding a match? (2) How many donors are needed to achieve a match probability of a given level?

Solution: Modeling matches as a sum of independent Bernoulli variables and using the CLT gives a match probability of about 0.86 for 200 000 donors. To reach the target probability the registry must contain roughly several hundred thousand donors, a very large number.

Example 6

From a large batch of seeds, 600 are sampled and 93 are found to be high quality. Find the confidence interval for the true high‑quality rate at a specified confidence level.

Solution: Let Xᵢ be the indicator that the i‑th seed is high quality. The sample proportion ̅p = 93/600. By the CLT, ̅p is approximately normal with variance ̅p(1‑̅p)/600, leading to the confidence interval … (numeric values omitted).

These examples illustrate that whenever a random variable can be expressed as a sum of independent and identically distributed components, the Central Limit Theorem enables probability calculations. In large‑sample settings it also underpins interval estimation and hypothesis testing for non‑normal populations, acting as a bridge between probability theory and mathematical statistics.

Hu Tiantui. “On the Central Limit Theorem in Mathematical Modeling.” Journal of Suzhou Vocational University, 2002(03):22‑24. DOI:10.16219/j.cnki.szxbzk.2002.03.007.