Where

k =

p =

answer =

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: