Computing Binomial Probabilities Using the Normal Curve

Beyond the empirical rule, we may apply the normal curve to approximating binomial probabilities . The key image is visualizing the binomial histogram as centered about np with SD= $\sqrt{np(1-p)}$. Here, for example, is the histogram of binomial probabilities for n=30 and p=.4.

  

Figure 4.7: Histogram of Probabilities for Binomial n=30 and p=.4

  

Figure: Approximating $P(X\leq 10)$ using Curve instead of Rectangles

The height of the rectangle over, say 10, is its probability P(X=10). However, since the width of the rectangle is 1, then

P(X=10)=height of rectangle over 10 = area of rectangle over 10.

What about $P(X\leq 10)$? This probability corresponds to the total area of the rectangles over and to the left of 10 (shaded rectangles in Figure  4.8).

Using the binomial probability function to compute the area (probability) of each shaded rectangle, the total area equals .2915. However, this requires repeated applications of the binomial formula (11 times, in fact). We may calculate a quick approximation of the desired probability by replacing the rectangles with a curve ! See shaded area under curve in Figure  4.8.

There are many normal curves, which one do we use to replace the rectangles? Answer: the curve which has the same mean and SD as the rectangles! The (binomial) rectangles have mean $\mu=n\;p=(30)(.4) = 12$ and SD $\sigma= \sqrt{np(1-p)}= \sqrt{ (30) (.4)(.6)} = 2.68$. Using the normal curve with the same mean and SD, the area to the left of 10.5 is .2878, which is a close estimate of the true area of .2915.

Similarly, P(X=14)=.1101 is approximated by .1124, the area under the normal curve between 13.5 and 14.5.