by using Theorem 1 of Central Limit Theorem (CLT),

computation

Normal Distribution parameters:

• 𝜇 = 𝐄[Binomial(𝑋)] = 𝑛𝑝
• 𝜎 = 𝑆𝑡𝑑(Binomial(𝑋)) = √[𝑛𝑝(1-𝑝)]

where:

• 0.05 ≤ 𝑝 ≤ 0.95
• 𝑛 is large

Info

In practice, the approximation is adequate provided that both 𝑛𝑝 ≥ 10 and 𝑛(1 − 𝑝) ≥ 10

Example

A new computer virus attacks a folder consisting of 200 files. Each file gets damaged with a probability of 0.2 independently of other files. What is the probability that fewer than 50 files get damaged

The number 𝑋 of damaged files has a Binomial distribution with:

• 𝑛 = 200
• 𝑝 = 0.2
• 𝜇 = 𝑛𝑝 = 40
• 𝜎 = √[𝑛𝑝(1 − 𝑝)] = 5.657

Applying the Central Limit Theorem with the continuity correction

Notice that the properly applied continuity correction replaces 50 with 49.5, not 50.5. Indeed, we are interested in the event that 𝑋 is strictly less than 50. This includes all values up to 49 and corresponds to the interval [0, 49] that we expand to [0, 49.5]. In other words, events {𝑋 < 50} and {𝑋 < 49.5} are the same; they include the same possible values of 𝑋. Events {𝑋 < 50} and {𝑋 < 50.5} are different because the former includes 𝑋 = 50, and the latter does not. Replacing {𝑋 < 50} with {𝑋 < 50.5} would have changed its probability and would have given a wrong answer