Distribution. Whenever the number of trials in a binomial experiment is small it is easy to find probabilities of the various values of x

Whenever the number of trials in a binomial experiment is small it is easy to find probabilities of the various values of x, the number of successes, by using formula

As the number of trials increases, however, the effort involved in answering questions about probabilities associated with the experiment quickly becomes laborious.

For instance, suppose that we want to know the probability that in fifteen tosses of a fair coin we toss at least nine heads. You will undoubtedly agree that n =15 is not a large number of trials. However, in order to find we say

P (x =9 or 10 or 11 or 12 or 13 or 14 or 15) =

So we have seven probabilities to compute, after which we must perform the addition. This is not practically difficult, but it takes a fair amount of time.

To find only one of these probabilities, for example, we have

;

.

Therefore

.

Thus, you see that if we were to calculate other six such probabilities we would expend a considerable amount of time and energy.

In such cases, the normal distribution can be used to approximate the binomial probability. Note that, for a binomial problem, the exact probability is obtained by using the binomial formula. If we apply the normal distribution to solve a binomial problem, the probability that we obtain is an approximation to the exact probability.

Example:

According to an estimate, 50% of the people have at least one credit card. If a random sample of 30 persons is taken, what is the probability 19 of them will have at least one credit card?

Solution:

Let n be the total number of persons in the sample, x be the number of persons in the sample who have at least one credit card, and p be the probability that a person has at least one credit card. Then, this is a binomial problem with

n =30; p =0.50; ;

x =19; and .

Using the binomial formula, the exact probability that 19 persons in a sample of 30 have at least one credit card is

Now let us solve this problem using the normal distribution as an approximation to the binomial distribution. For this example,

and

.

Using the normal distribution as an approximation to the binomial involves the following steps:

Step1:

Compute and for the binomial distribution.

To use the normal distribution, we need to know the mean and standard deviation of the distribution. Hence, the first step in using the normal approximation to the binomial distribution is to compute the mean and standard deviation of the binomial distribution. As we know the mean and standard deviation of the binomial distribution are given by

and .

Using these formulas, we obtain

;

.

Step2:

Convert the discrete random variable to a continuous random variable.

The normal distribution applies to a continuous random variable, whereas the binomial distribution applies to a discrete random variable. The second step is to convert the discrete random variable to a continuous random variable by making the correction for continuity.

To make the correction for continuity, we use the interval 18.5 to 19.5 for 19 persons.

Step3:

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