Binomial Theorem > Success/Failure Condition

## What is the Success/Failure Condition?

Sometimes with a binomial experiment, we can use a normal distribution to approximate a binomial; This can make finding probabilities easier for some experiments—especially when you’re dealing with large samples. The Success/Failure condition is used to figure out if a sample size in a binomial experiment is large enough to use the normal approximation. It could more appropriately be called the**approximately normal condition**, and here’s why.

As a sample size increases, we can use the

**normal distribution to approximate a binomial**. The question is, how large must n be, before we can use the normal distribution? The success/failure condition gives us the answer:

Success/Failure Condition: if we have 5 or more successes in a binomial experiment (n*p ≥ 10)

and5 or more failures (n*q ≥ 10), then you can use a normal distribution to approximate a binomial (some texts put this figure at 10).

Where:

- n = the sample size.
- p = the probability of success.
- q = the probability of failure (this can also be written as 1 – p).

## Example Problem

**Example problem:** Sixty two percent of part time employees in a certain city are receiving SNAP benefits (food stamps). Check the success/failure condition if a sample of 500 part time employees are selected.

Step 1: Find n, p, and q:

- n, the sample size, is 500,
- p, the probability of “success” (in this case, the probability of someone being on food stamps) is 62%, or 0.62,
- q, the probability of “failure” is 1 – p = 1 – 0.62 = 0.38.

Step 2: Figure out if you can use the normal distribution:

- n * p = 500 * .62 = 310 and,
- n * q = 500 * .38 = 190.

These are both larger than 5, so the success/failure condition is met.

**Notes**:

- If you are dealing with probabilities two proportions, both samples must meet the condition in order to use it.
- As the condition can differ with the lower values (e.g. np=5 or np =10), check with your textbook author or professor to see what values they prefer before using the condition.

## References

De Veaux, R. (2005). Intro Stats (2nd Edition). Addison-Wesley.