The Binomial Distribution

Two Experiments

During the last class (so long ago) we discussed the probability distributions for two experiments:

• Guessing the answers for a 3 question quiz where each question had 5 options, and
• Counting the number of hits that Buster Posey gets in a randomly selected game.

Let’s discuss the ways that these are similar and different.

Binomial Experiments

A binomial experiment is one where there are:

• \(n\) independent trials.
• Each trial has only two outcomes. (Success or Failure).
• The probability of success in any trial is \(p\) (failure is \(q\)).
• You want to compute the probability of \(x\) successes.

Binomial Probability Formula

The pdf for a binomial experiment is: \[ p(x) = ( _{n}C_{x} )\ p^{x}\ q^{n-x} \] Let’s have a quick discussion of why it looks like this.

Passing the class. (using the formula)

About 62% of students who take Math 200 pass. If a class has 52 students what is the probability that:

• exactly 30 pass,
• exactly 25 pass, or
• exactly 40 pass.

Looking at the disribution (simulation)

Here is a quick simulation of

A different type of problem

Suppose we are back to guessing on our little multiple choice test. (3 questions, 5 options each) Then what is the probability that you get:

• less than 2 correct, or
• at least 2 correct.

Is there a quicker way?

There are two calculator commands that make this faster:

- binompdf(n,p,x)
- binomcdf(n,p,x)

The question is how do they differ and how do we use them?

Example

CBS televised a recent Super Bowl football game between the New Orleans Saints and the Indianapolis Colts. That game received a rating of 45, indicating that among all U.S. households, 45% were tuned to the game (based on data from Nielsen Media Research). An advertiser wants to obtain a second opinion by conducting its own survey, and a pilot survey begins with 12 randomly selected households.

• Find the probability that none of the households is tuned to the Saints/Colts game.
• Find the probability that at most one household is tuned to the Saints/Colts game.
• Find the probability that at least one household is tuned to the Saints/Colts game.
• If at most one household is tuned to the Saints/Colts game, does it appear that the 45 share value is wrong? Why or why not?

Example

Based on the American Chemical Society, there is a 0.9 probability that in the United States, a randomly selected dollar bill is tainted with traces of cocaine. Assume that eight dollar bills are randomly selected.

• Find the probability that all of them have traces of cocaine.
• Find the probability that exactly seven of them have traces of cocaine.
• Find the probability that the number of dollar bills with traces of cocaine is seven or more.
• If we randomly select eight dollar bills, is seven an unusually high number for those with traces of cocaine?

Example

Based on a Harris Interactive poll, 20% of adults believe in reincarnation. Assume that six adults are randomly selected, and find the indicated probability.

• What is the probability that exactly five of the selected adults believe in reincarnation?
• What is the probability that all of the selected adults believe in reincarnation?
• What is the probability that at least five of the selected adults believe in reincarnation?
• If six adults are randomly selected, is five an unusually high number who believe in reincarnation?