Download CN 8.1B The Binomial Theorem - Fort Thomas Independent Schools

AP Statistics
Notes
Name: ____________
Date: _____________
Lesson 8.1B: The Binomial Probability Theorem
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 coefficient
1.
Binomial probability formula (binomial pdf and cdf)
Binomial Probability Theorem
Recall the example from the previous lesson:
Ex:
Brighton Eager wants to determine the probability that he passes the MC
portion of an AP Stats test that has 10 MC questions, each with answer
choices A-E, given that he randomly guesses on each question.
What is the probability that Brighton answers at least 6 of the 10 MC
questions correctly, thus earning at least a 60% (passing grade) on that
portion of the test. In statistical notation, find P( X  6) .
Let’s look at a simpler problem first – the probability that Brighton answered
exactly 6 of the 10 questions correctly  P( X  6) .
Step 1: Find the probability of getting 6 successes in 10 observations in a
specific case, say, SSSSSSFFFF.
Step 2: Determine how many arrangements there are for 6 successes in 10
observations. [Hint: Think combinations -- n Ck !]
Step 3: Find P(X = 6)
P(X = k)  binompdf(n, p, k)
The Binomial Probability Theorem
If X has the binomial distribution with n observations and probability p of
success on each observation, the possible values of X are 0, 1, 2, … n.
If k is any one of these values,
n
P(X = k) =   p k (1-p) nk
k 
n
The expression   is called the binomial coefficient.
k 
n
n!
  =
k!(n  k )!
k 
As a class, let’s find P(X  6), the probability that Brighton passes the MC part.
P(X  k)  binomcdf(n, p, k)
2.
Binomial Probability Distribution
Let’s have the calculator generate the probability distribution for the binomial
random variable, X, in our Brighton Eager example.
1. Store the values 0, 1, 2, … 10 in L 1 .
Seq(x, x, 0, 10, 1)
2. Store the probabilities in L 2 .
binompdf(10, .20, L 1 )  L 2
3. Construct a probability histogram
using the values in L 1 and their
probabilities in L 2 .
StatPlot
4. Store cumulative probabilities in L 3 .
binomcdf(10, .20, L 1 )  L 3
5. Construct a cumulative probability
histogram using the values in L 1
and their cumulative probabilities in L 3 .
StatPlot
List/Ops
AP Statistics
Notes
Name: ____________
Date: _____________
Lesson 8.1B: Permutations, Combinations, Pascal’s Triangle
Suppose that Amy, Brad, Carla, and Danny are the four members of a small club here
at Highlands High School. They decide to elect a president, secretary, and treasurer for
the club, with no one person holding more than one position. In how many ways can
they do this? List all of the possible outcomes.
Now suppose that these four people need to select 3 members to act as representatives
on a school council. In how many ways can they do this? List all possible outcomes.
Suppose that Amy, Brad, Carla, and Danny decide that they don’t really need a
secretary – just a president and a treasurer. In how many ways can they do this? List
all of the possible outcomes.
Suppose that they need to select 2 members to talk to Mr. Robinson about some
fundraising events. In how many ways can they do this? List all possible outcomes.
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