Binomial Probability Definition
The Binomial distribution is used for outcome variables with two exclusive categories, such as "Yes" and "No", "Success" and "Failure". One common example is the coin toss, with two exclusive outcomes of "head" and "tail".
Binomial Probability Examples
For a pickup soccer game with 12 people signed up, we can assume that the probability of each person showing up is at 0.9. The "exact probability" that we get EXACTLY 9 people is at around 0.085; the "cumulative probability" that we get NO MORE THAN 9 people is at 0.11, and the probability that we get more than 9 people (10 people, or 11 people, or 12 people) is at 1-0.11=0.89.
For a fair coin toss, the probability of getting a head or a tail is each at 0.5. If we toss a fair coin 10 times, the "exact probability" that we get EXACTLY 4 heads is at 0.205078125; and the "cumulative probability" that we get NO MORE THAN 4 heads (0 heads, or 1 head, or 2 heads, or 3 heads or 4 heads) is at 0.376953125.
In Microsoft Excel, BINOM.DIST(4,10,0.5,0)=0.205078125; BINOM.DIST(4,10,0.5,1)=0.376953125
Similarly in TI83/TI84, binompdf(10,0.5,4)=0.205078125; binomcdf(10,0.5,4)=0.376953125