
Latin
Latin/Greek Root Words


(Statistics
connection) 

Chap. 6
Binomial and Geometric Distributions
AP Statistics Standards
III. Anticipating Patterns:
(continued)
A. Probability

Interpreting
probability, including longrun relative frequency
interpretation

“Law of Large Numbers”
concept

Discrete random
variables and their probability distributions, including
binomial and geometric

Simulation of random
behavior and probability distributions

Mean (expected value)
and standard deviation of a random variable, and linear
transformation of a random variable
B. Combining independent
random variables

Notion of independence
versus dependence

Mean and standard deviation for sums and differences of
independent random
variables

Objectives 
Essential Question:
Can humans simulate a random
process and why is this an important issue? 
Random Variables

Be as one with the
following vocabulary:
discrete distribution: A
density curve (theoretical model of a probability distribution)
that has a finite number of possible
values within any finite segment of its range of
values. Discrete distributions tend to
look like they are made of stair steps. Examples include the
binomial and geometric distributions.
continuous
distribution: A density curve (theoretical model of a
probability distribution) that has an
infinite number of possible values within any finite
segment of its range of values.
Continuous distributions form smoothlooking curves. Examples
include the normal, student t, and chi squared distributions.

Plot discrete probability distributions for
simple systems such as flipping coins.

Be as one with the law of large numbers.
Average results
of many independent observations are stable and predictable

Describe the law of small numbers.
The tendency to draw unreliable
inferences based on a small number of observations.
Relevance:
The law of small numbers is a major factor in all kinds of
misconceptions and maladies such as hothand, lucky charms, and
compulsive gambling behavior.

Find the mean of a
discrete probability distribution. The
mean is a weighted average.
μ_{x} = ∑ (x_{i})^{ }P_{i}
Homefun
(formative/summative assessment): 
Read section 6.1, Exercises 1, 5, 7 pp. 353 to 355


Stats
Investigation:
Simulation of a Random Process ( Teams of two) 
Purpose:
Determine if actual results of flipping a coin match results
simulated by humans.
Instructions: Each
partner in a two person team will record 100 fake coin toss
tosses on a sheet of paper using H for heads and T for Tails.
The team will then record 200 actual tosses. The experimenters
will then circle the runs of 2 or more heads within the data
both for the fake and real tosses. For example the following
data contains 2 runs of 3, 1 run of 6, and 3 runs of 2 heads:
HHHTTHHHTHTHTTTHHHHHHTTTTHHTTTTHHTHH
The experimenters will post the
number of runs of each length on the board. When all results
are posted for the class, the number of runs vs. size of runs
is to be plotted for the classes fake and real data ( two
separate plots).
Questions /Conclusions:
Answer with short paragraphs.
 What is the biggest
difference between the real and fake data. Speculate about
why this is the way it is.
 Are humans reliable at generating random numbers?
Discuss this both in terms of the law of large numbers and
the law of small numbers (p. 392)
Resources/Materials: pennies
for flipping 

Essential Question:
Can standard deviations be added? 
Random Variables

Calculate the
standard deviation of a discrete random variable. (p.410).
Var = ∑ (x_{i}
 μ_{x})^{2 }P_{i}
example problem

Apply the
rules for means.

if the units
of each data point are changed, then the units of the mean are also
changed in the same way

The mean of the sum of random variables is the same as the sum of their means.

Apply the
rules for the variance of a distribution in which multiple random variables are combined.

The random variables must be independent

The variances are additive even when taking the difference between two random variables.

Standard deviations are not additive.
Homefun
(formative/summative assessment): Read 6.2
Exercises 13, 17, 27, 29, 31, 32, 33, 34
pp. 354 to 357

Essential Question: What will changing the units of
measurement do to measures of spread and central tendency? 
Linear Transforms
 What is a linear transform? (See p. 53).
x_{new} = a + b∙x_{old}
Example: the linear transform
to change from Celsius to Fahrenheit
( ºF ) = 32 + 1.8 ( ºC )
 State the effect that multiplying each number by
a constant and/or adding a
constant to each number in a data set has on the following: (This effect is important to
know when changing the data points' units.)

(each data
point) + a 
(each data
point) * b 
mean 
(mean) + a 
(mean) * b 
median 
(median) + a 
(median) * b 
standard deviation 
no change 
(std dev) * b 
IQR 
no change 
(IQR) * b 
range 
no change 
(range) * b 
Q1 and Q3 
(Q1) + a & (Q3) + a 
(Q1) * b & (Q3) * b 
Formative assessment: Using the TI83 calculators and monthly temperature data in Fahrenheit from the GeenvilleSpartenburg Airport, calculate and record the mean, median, standard
deviation, IQR, range, Q1, and Q3 for both Fahrenheit and Celsius. Convert the recorded Fahrenheit values directly to Celsius. How do the converted values compare to the ones calculated in Celsius.?
Homefun (formative/summative assessment): Exercise 41, 47, 49 pp. 378 to 379 

Essential Question:
Do we live in a binary world? 
 The Binomial Distribution
 Relevance: The
binomial distribution is frequently used for analyzing and setting
up surveys.
 Be as one with
the Binomial Setting, SNIP.
Success or failureobservations are divided into these categories.
Number of observations is fixed.
Independent observations
Probability of success is constant
 Use the binomial coefficient or
combinations.
The
number of ways of arranging k successes among n observations.
Note: 0!
= 1

Calculate binomial probabilities (p. 448).
P(X=k) 
= 
( 
n 
) 
p^{k} (1  p) ^{
(nk)} 
k 

Calculate combinations and binomial distributions with a TI
 83
 Binomial Distributions:
2nd
VARS binompdf
or cdf (no,
pigs,
killed).
binomepdf
: 
finds
the probability for a single value of k. Think
p
stands for
particular. 
binomecdf
: 
finds
the probability for the cumulative
values from 0 to k.

Formative assessment: create the 3coins distribution using the TI83 calculator.
 Calculate means and standard deviations
for the binomial distr.
(Click
here to see an example
of count vs fraction type data for the 3 coins example.)
Data Type 
Mean 
Std Dev 
count 
np 
[ np (1  p) ]^0.5 
fraction

p 
[ p (1  p) / n ]^0.5 
Homefun
(formative/summative assessment): Read 6.3 Exercises 73, 75, 81, 83, 87, 93 pp. 403 to 405 


Essential Question:
When is estimating how many tries
before success an issue? 
The Geometric Distribution
Relevance: The
geometric distribution used for analyzing the probability of an
even occurring for the first time, such as the probability of a
baseball player getting a hit for the first time vs. the number of
times at bat.

Be aware of the key
differences between binomial and geometric distributions.

Binomial:

Finds the
probability that k success will occur in n
number of attempts.
Geometric:
Finds the probability that
a success will occur for the first time on the nth try.
equation:
P(x=n)
= ( 1  p )^{n1} p
 Be as one with
the Geometric Setting on page 464,
SPIT
Success or failureobservations are divided into these categories.
Probability of success is constant
Independent observations
Trials the number of trials until first success is the goal.
 Correctly use the geometric distribution
for calculating probabilities with the TI83 calculator.
2nd
VARS
geomepdf
or geomecdf
(pig,
xing)
geomepdf
: 
finds the
probability for a single value of x. Think
p
stands for
particular. 
geomecdf
: 
finds the
probability for the cumulative
values from 1 to x.

Formative assessment: create the distribution for the probability of hitting the baseball for the 1st time vs number of times at bat using the TI83 calculator and a batting average of 300 (30% chance of hitting the ball).
Homefun
(formative/summative assessment): Exercises 95, 97, 101, 105 pp. 405
to 406
Essential Question:
How can I make an
"A" on the test? 
Binomial and Geometric Distribution
Review
 Master the vocabulary
 Work the practice test.
 Review the objectives.
Homefun (formative/summative assessment): Chapter 6 AP Statistics Practice Test pp.409 to 411
Summative Assessment:
Unit Exam objectives 1 18


