Combinations are the way to count the number of ways that x things can be chosen from n. These are needed to determine the probability that a binomial random variable takes on a particular value.

Probabilities for binomial random variables can be determined by formula or by using a table in the back of the book. You should know both.

Every binomial random variable is described by two numbers, or parameters. n is the number of trials, and p is the probability of a success.

Since the number of Duquesne students is very large compared to our sample size, we can model this random sampling by taking a box with 6 black balls (for the females) and 4 white balls (for the males) and taking a sample of size three, one at a time, replacing the drawn ball each time.

Here is a list of all the possible sequences of outcomes.

WWW which has probability (.4)(.4)(.4) = (.4)^3 WWB which has probability (.4)(.4)(.6) = (.4)^2 (.6) WBW which has probability (.4)(.6)(.4) = (.4)^2 (.6) BWW which has probability (.6)(.4)(.4) = (.4)^2 (.6) WBB which has probability (.4)(.6)(.6) = (.4) (.6)^2 BWB which has probability (.6)(.4)(.6) = (.4) (.6)^2 BBW which has probability (.6)(.6)(.4) = (.4) (.6)^2 BBB which has probability (.6)(.6)(.6) = (.6)^3There are three sequences with exactly two black balls, so the probability is 3 (.4) (.6)^2.

The binomial distribution is appropriate when we have this setting:

- there are a fixed number of Bernoulli trials;
- there are two possible outcomes for each trial;
- the trials are independent of one another;
- there is the same chance of success for each trial;

nCx = n! / (x! (n-x)! )The notation ! is called "factorial". For example,

5! = 5*4*3*2*1 = 120 4! = 4*3*2*1 = 24 3! = 3*2*1 = 6 2! = 2*1 = 2 1! = 1 = 1 0! = 10! is defined as 1 so that the formula nCx makes sense when x=0 or x=n.

Here are some example calculations:

3C2 = 3! / (2! 1!) = (6 / 2) = 3 5C2 = 5! / (2! 3!) = (5*4) / (2*1) = 10When doing by hand, it is generally best to cancel as much as you can first. Many of you can do this calculation on your calculators directly.

P( X = x ) = nCx p^x (1-p)^(n-x) for x = 0, 1, 2, ..., nShow how this formula holds for the example.

Here are a few examples:

**Example:**
n = 15, p = .4.

- Find P(X = 10).
- Find P(X < 10).
- Find P(3 < X < 10).
- Find P(X > 10).

- P(X = 10) = P(X <= 10) - P(X <= 9) = .9907 - .9662 = .0245
- P(X < 10) = P(X <= 9) = .9662
- P(3 < X < 10) = P(X <= 9) - P(X <= 3) = .9662 - .0905 = .8757
- P(X > 10) = 1 - P(X <= 10) = .9907

np = mu is the mean of the distribution.

sqrt(n*p*(1-p)) = sigma is the standard deviation of the distribution.

Last modified: Feb 5, 1996

Bret Larget, larget@mathcs.duq.edu