AP Statistics
Notes
Name: ____________
Date: _____________
Lesson 8.1A: The Binomial Distribution
Learning Targets:
A: Identify a random variable as binomial by verifying four conditions: two outcomes
(success and failure); fixed number of trials; independent trials; and the same
probability of success for each trial.
B: Use technology or the formula to determine binomial probabilities and to construct
probability distribution tables and histograms.
C: Calculate cumulative distribution functions for binomial random variables and
construct cumulative distribution tables and histograms.
Vocabulary:
Binomial setting (Four conditions)
1.
Binomial random variable
Characteristics of the Binomial Setting
In this lesson, we will be studying situations in which there are two outcomes of
interest.
 Flipping a coin
Head, Tail
 Having a child
Boy, Girl
 Selecting an item from
a production process
Defective, Not Defective
Ex:
Brighton Eager (pronounced “Bright and Eager”) is extremely concerned
about his performance on the multiple-choice questions on AP Stats tests.
As you know, each chapter test contains 10 multiple-choice questions,
each with choices A through E. Unfortunately, Brighton does not pay
attention in class nor does he complete his homework. To top it off, he
never studies for a test! Thus on test day, he has no choice but to
randomly guess on each of the 10 multiple-choice questions. Brighton
would like to know the probability that he answers at least 6 of the 10
multiple-choice questions correctly, thus earning at least a 60% (passing
grade) on that portion of the test.
Verify that the scenario with Brighton Eager and his MC test questions is a
binomial setting in that it meets the four characteristics listed in the table below.
1.
Each observation falls into one of two categories, called Success and Failure.
2.
There is a fixed number, n, of observations.
3.
The n observations are all independent, meaning any one observation has no
influence on any other observation.
4.
The probability of Success, called p, is the same for each observation.
Just for a minute, pretend that you’re in Brighton Eager’s shoes.
1. Randomly fill in a bubble for each of the 10 questions
on the provided Scan Tron sheet.
2. When given the answer key, grade your quiz.
Number correct = ____________
3. What is P(Answer any given question correctly)?
4. How many questions would you expect to get correct?
5. Is it very likely that you’d get all 10 questions correct?
2.
The Binomial Random Variable
Let the random variable, X, be the number of Successes in a binomial setting.
Then X is called a binomial random variable with parameters n and p  B(n,p)
where n is the number of observations and p is the probability of a success on
any one observation. The possible values of X are 0 to n.
For Brighton Eager, X = __________________________ and B(n,p) = ____________
3.
Binomial Probabilities
Let’s help Brighton Eager determine the probability that he passes the MC
portion of an AP Stats test by looking at the probabilities that a binomial random
variable, X, takes on any of its values 0, 1, 2, … n. Specifically we want to know
P( X  6) . We will estimate this probability by using a simulation.
Describe how you can simulate taking 20 quizzes on which you randomly guess
on each of 10 questions with answer choices A-E.
Keep track of how many correct answers you get per quiz and tally your results below.
# Correct
0
1
2
3
4
5
6
7
8
9
Your Tally
Class Tally
How many times did you “pass” the quiz? ______ What is your “pass rate”? ______
Record your results on the board so that we can get a class “pass rate.”
Total # Pass
 ___________ 
Total # Simulations
10