Do Now
*Before you begin, take out your homework!
• A group of people are concerned that the coach of a local high school
men’s and women’s basketball teams alters the amount of air in the
basketball to gain an unfair advantage over opponents during home
games. The idea is that the basketballs are pumped up with one pound
per square inch less air than required, and his teams practiced with these
altered balls all week prior to home basketball games. Since these underpumped basketballs would react differently to being shot at a basket, the
team that practiced with these balls would have an unfair advantage when
it came to shooting free throws.
• Create a diagram to design this experiment.
• A statistics teacher wants to know how her students feel about an
introductory statistics course. She decides to administer a survey to a
random sample of students taking the course. She already knows that
there's an innate difference in how the males and females feel about the
class.
• In this situation, what would be the best sampling method for this teacher to use
Agenda
•
•
•
•
•
•
Review Do Now/HW
Announcements
Overview of Unit 4
Review Do Now/Homework
Introduction to Probability
Lesson on Simulating Events
Announcements!
• Ms. Matrone will not be here on Thursday –
therefore…SURPRISE! Your Unit 3 test will be
on Thursday instead.
….and yes, today we are learning something
new!
PS: Corrections are also due to the sub on
Thursday! Don’t forget!
Unit 4 Overview
• Decide if a specified model is consistent with
results from a given data-generating process, e.g.,
using simulation.
• Understand independence and conditional
probability and use them to interpret data.
• Use the rules of probability to compute
probabilities of compound events in a uniform
probability model.
• Use probability to evaluate outcomes of
decisions.
Probability
• The probability of any outcome of a chance
process is a number between 0 (never occurs)
and 1(always occurs) that describes the
proportion of times the outcome would occur
in a very long series of repetitions.
The Idea of Probability
• Chance behavior is unpredictable in the short
run, but has a regular and predictable pattern
in the long run.
• The law of large numbers says that if we
observe more and more repetitions of any
chance process, the proportion of times that a
specific outcome occurs approaches a single
value.
Probability helps us answer questions
such as….
• If we know the blood types of a man a
woman, what can we say about the blood
types of their future children?
• Give a test for the AIDs virus to the employees
of a small company. What is the chance of at
least one positive test if all the people test are
free of the virus?
Three Methods for Answer Questions
Involving Chance
• Actually observing the random phenomena many
times and calculating the frequency (percentages) of
the results – Slow and sometimes expensive and often
impractical.
• Developing a probability model and using it to
calculate a theoretical answer – Requires we know
something about the rules of probability (which we
have not learned yet)
• Starting with a model that reflects the truth about a
random phenomenon and then develop a plan to
simulate the phenomenon as if it actually occurred–
Quicker than repeating the real procedure. Allows us to
get reasonably accurate results.
Simulation
• The imitation of chance behavior, based on a
model that accurately reflects the experiment
under consideration
• Simulation refers to the imitation or
representation of a real life situation.
• The imitation of chance behavior, based on a
model that accurately reflects the situation, is
called a simulation.
Why Do we Simulate Events?
• Quicker than repeating the real procedure.
• Allows us to get reasonably accurate results.
• Inexpensive, doesn’t require many resources
We can use physical devices,
random numbers, and technology
to perform simulations!
Steps in Simulation
1. Describe the problem, defining key components.
2. List assumptions about the problem.
3. State what one trial would be, including a
stopping rule if necessary
4. Assign numbers to represent the outcomes
5. Perform simulation by conducting trials and
recording observations (AP Exam Requires you to
use a random digit table)
6. Summarize and State conclusions (only
appropriate to do if you have at least 100 trials)
Example 1
Michael Jordan had a 90% free throw
percentage when he came to the free throw
line. Set up a simulation for finding the
probability that Michael would make the
next 6 straight free throws.
Problem
• Find the probability that Michael makes 6 free
throws in a row.
Assumptions
• Each shot is independent of the next and any
shot has an equally likely chance of being
either a missed shot or a made shot. Thus
each digit we assign is independent of the
next.
Explain One Trial
• One trial represents 6 free throws (or 6
random digits)
Assignment of Digits
• Since Michael makes 90% of the shots he
takes, we will assign 90% of our digits (9
digits) to shots he makes. 1-9 will represent
shots Michael makes. Since Michael misses
shots 10% of the time, we will have 10% of
digits (1 digit) assigned to a miss. 0 will
represent a miss.
Simulation
• Since we are looking at the probability that he
makes 6 free throws, look at 6 random
numbers at a time. Tally how many times 6
numbers in a row exclude 0. Run the
simulation for 50 trials by looking at 6 single
digit numbers for 50 groups of 6.
Simulation Ctd.
Conclusions
• Conclusions: Determine out of the 20 trials
how many sets of 6 have no #9 (no miss). This
is the probability required.
Example 2
Assuming the same information from example 1
determine the probability that Michael will
make 5 baskets in a row and miss the 6th.
Problem
• Find the probability that Michael makes 5 free
throws in a row then misses the 6th free throw
Assumptions
• Digits 0-9 are equally likely and independent
of one another
Explain One Trial
• One trial is equivalent to 6 free throws (which
is represented by 6 random digits.)
Assign Digits
• 1-9 represents Michael hits the free throw. 0
represents that he misses.
• 0-8 represents Michael hits the free throw. 9
represents that he misses
Either of these could work!
Simulation
• Look at 6 random numbers at a time.
• Determine if the first 5 digits have no 0’s
• If all five have no 0’s then look at the next
number (6th number) to determine if it is a 0
or not.
• Tally how many times this event occurs out of
20 trials.
Conclusions
• Draw conclusions about the probability.
Example 3
• Suppose every couple continues having children
until they have a boy and then they stop. No
couple can have more than one boy while they
may have several girls. Does this mean that the
population will become overwhelmingly female?
• Design a simulation and test it using the random
number table.
– THINK… What is the probability that a couple will
have a girl, what is the probability that a couple will
have a boy? Think about this before assigning your
digits!
Possible Solution
• Problem: Each couple has children until a boy occurs.
Find the probability of males and females out of a
certain number of trials
• Assume: even and odd digits between 0-9 are
equally likely and even digits and odd digits are
independent of one another
• Assign digits: 0-9 even digits represent having
a boy 0-9 odd digits represent having a girl.
• Simulation: Look at random numbers recording the
results until we obtain one even number. The
number of digits included in this one trial represents
one family.
Possible Solution (Continued)
• Draw conclusions: Find the number of females in the
families. Determine the probabilities out of the total
families looked at. Compare probabilities of males
and state conclusions.
– Total of 144 boys and 156 girls
– 48% boys and 52% girls is not greatly different from the
expected 50% for each gender
– The reasoning that the population will be overwhelmingly
female is not taking into account the large number of
families that have no girls.
Classwork Problems
• Pg. 398 #6.3, 6.4,6.5
• Homework: Finish this & independent practice
and STUDY FOR TEST!