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The normal approximation to the binomial is a probability distribution method used to approximate the binomial probability distribution when the number of trials is large and the probability of success is close to 0.5. An example of this would be tossing a fair coin 10 times and trying to determine the probability of getting 5 heads. The normal approximation to the binomial can be used to approximate this probability.
If X is a that follows a with n trials and p probability of success on a given trial, then we can calculate the mean (μ) and standard deviation (σ) of X using the following formulas:
- μ = np
- σ = √np(1-p)
It turns out that if n is sufficiently large then we can actually use the to approximate the probabilities related to the binomial distribution. This is known as the normal approximation to the binomial.
For n to be “sufficiently large” it needs to meet the following criteria:
- np ≥ 5
- n(1-p) ≥ 5
When both criteria are met, we can use the normal distribution to answer probability questions related to the binomial distribution.
However, the normal distribution is a continuous probability distribution while the binomial distribution is a discrete probability distribution, so we must apply a continuity correction when calculating probabilities.
In simple terms, a continuity correction is the name given to adding or subtracting 0.5 to a discrete x-value.
For example, suppose we would like to find the probability that a coin lands on heads less than or equal to 45 times during 100 flips. That is, we want to find P(X ≤ 45). To use the normal distribution to approximate the binomial distribution, we would instead find P(X ≤ 45.5).
The following table shows when you should add or subtract 0.5, based on the type of probability you’re trying to find:
| Using Binomial Distribution | Using Normal Distribution with Continuity Correction |
|---|---|
| X = 45 | 44.5 < X < 45.5 |
| X ≤ 45 | X < 45.5 |
| X < 45 | X < 44.5 |
| X ≥ 45 | X > 44.5 |
| X > 45 | X > 45.5 |
The following step-by-step example shows how to use the normal distribution to approximate the binomial distribution.
Example: Normal Approximation to the Binomial
Suppose we want to know the probability that a coin lands on heads less than or equal to 43 times during 100 flips.
In this situation we have the following values:
- n (number of trials) = 100
- X (number of successes) = 43
- p (probability of success on a given trial) = 0.50
To calculate the probability of the coin landing on heads less than or equal to 43 times, we can use the following steps:
First, we must verify that the following criteria are met:
- np ≥ 5
- n(1-p) ≥ 5
In this case, we have:
- np = 100*0.5 = 50
- n(1-p) = 100*(1 – 0.5) = 100*0.5 = 50
Both numbers are greater than 5, so we’re safe to use the normal approximation.
Step 2: Determine the continuity correction to apply.
Referring to the table above, we see that we should add 0.5 when we’re working with a probability in the form of X ≤ 43. Thus, we will be finding P(X< 43.5).
Step 3: Find the mean (μ) and standard deviation (σ) of the binomial distribution.
μ = n*p = 100*0.5 = 50
σ = √n*p*(1-p) = √100*.5*(1-.5) = √25 = 5
Step 4: Find the z-score using the mean and standard deviation found in the previous step.
z = (x – μ) / σ = (43.5 – 50) / 5 = -6.5 / 5 = -1.3.
Step 5: Find the probability associated with the z-score.
We can use the to find that the area under the standard normal curve to the left of -1.3 is .0968.
Thus, the probability that a coin lands on heads less than or equal to 43 times during 100 flips is .0968.
This example illustrated the following:
- We had a situation where a random variable followed a binomial distribution.
- We wanted to find the probability of obtaining a certain value for this random variable.
- Since the sample size (n = 100 trials) was sufficiently large, we were able to use the normal distribution to approximate the binomial distribution.
This is a complete example of how to use the normal approximation to find probabilities related to the binomial distribution.