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where
✓ n is the fixed number of trials.
✓ x is the specified number of successes.
✓ n – x is the number of failures.
✓ p is the probability of success on any given trial.
✓ 1 – p is the probability of failure on any given trial. (Note: Some textbooks
use the letter q to denote the probability of failure rather than 1 – p.)
These probabilities hold for any value of X between 0 (lowest number of pos-
sible successes in n trials) and n (highest number of possible successes).

The number of ways to rearrange x successes among n trials is called “n
choose x,” and the notation is
. It’s important to note that this math
Part III: Distributions and the Central Limit Theorem
expression is not a fraction; it’s math shorthand to represent the number of
ways to do these types of rearrangements.
In general, to calculate “n choose x,” you use the following formula:
The notation n! stands for n-factorial, the number of ways to rearrange n
items. To calculate n!, you multiply n(n – 1)(n – 2) . . . (2)(1). For example 5! is
5(4)(3)(2)(1) = 120; 2! is 2(1) = 2; and 1! is 1. By convention, 0! equals 1.
Suppose you have to cross three traffic lights on your way to work. Let X be
the number of red lights you hit out of the three. How many ways can you hit
two red lights on your way to work? Well, you could hit a green one first, then
the other two red; or you could hit the green one in the middle and have red
ones for the first and third lights, or you could hit red first, then another red,
then green. Letting G = green and R=red, you can write these three possibili-
ties as: GRR, RGR, RRG. So you can hit two red lights on your way to work in
three ways, right?
Check the math. In this example, a “trial” is a traffic light; and a “success”
is a red light. (I know, that seems weird, but a success is whatever you are
interested in counting, good or bad.) So you have n = 3 total traffic lights, and
you’re interested in the situation where you get x = 2 red ones. Using the

fancy notation, means “3 choose 2” and stands for the number of ways
to rearrange 2 successes in 3 trials.

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