- Use normal probability distribution to calculate binomial probabilities
- Check the conditions for applying a normal distribution to approximate a binomial distribution
The Connection between Binomial and Normal Distributions
Is it large enough?
Previously, we learned about the role that [latex]p[/latex] plays in the shape of the binomial distribution. The binomial probability distribution is skewed right if [latex]p < 0.5[/latex], symmetric and approximately bell shaped if [latex]p= 0.5[/latex], and skewed left if [latex]p > 0.5[/latex].
Let’s discuss the role that [latex]n[/latex] plays in its shape. For a fixed [latex]p[/latex], as the number of trials, [latex]n[/latex], in a binomial experiment increases, the probability distribution of the random variable [latex]X[/latex] becomes nearly symmetric and bell shaped.
condition for a normal distribution
The binomial distribution can be approximated well by the normal distribution when n is large enough so that the expected number of successes, [latex]np[/latex], and the expected number of failures, [latex]n(1-p)[/latex], are both at least [latex]10[/latex].
That is: The probability distribution will be approximately symmetric and bell shaped if
- [latex]np \geq 10[/latex] AND
- [latex]n(1-p) \geq 10[/latex]