Binomial Probabilities and Their Estimation

To compute the probability of a certain number of outcomes (k) of a certain number of attempts (n), given that the probability of that outcome is p, is:

nCk * pk * (1-p)(n - k)

Where nCk means the number of combinations of n taken k at a time. This is:
n!
k! (n - k)!

Using Javascript to Compute

It is easy to compute a small number of possible outcomes by adding together the probability of each. Type the values of n, k and p:
n =
k =
p =

answer =
When the number of outcomes gets large, however (like the probability of 10,000 or more rejects out of every million widgets manufactured), then we overly tax the power of the computer, when it is easy enough to estimate the probability.

Since the average number of occurances (µ=np) is distributed normally with a variance of np(1-p) (the standard deviation is the square root of the variance, of course), we can compute a z-score, then use a table to Look up the area under the Normal (Guasssian) Distribution. (this estimate should only be used for when the expected occurances and the expected "non-occurances" are both greater than 10 (that is, np > 10 and n(1-p) > 10)

Since we are moving from a discrete histogram-like distribution (plotting the frequency of outcomes, which are always an integer) to a continuous Normal Curve, we make an adjustment to include the area in between. So to compute 3 or more, we take the area of the curve to the right of the z-score of 2.5. To computer the area of 3 or less we would examine the area to the left of the z-score of 3.5.

Here are the z-scores to estimate for the n, k and p values above:
z-score for k or more:
z-score for k or less:

See if you can use your web page to answer question 25 of the Chapter 4 Quiz