Download Unit 9 2013-14 - Youngstown City Schools

YOUNGSTOWN CITY SCHOOLS
MATH: PRECALCULUS
UNIT 9: PROBABLILITY, RANDOM VARIABLES AND EXPECTED VALUE (3 WEEKS) 2013-2014
Synopsis: Students will work with random variables finding probability distributions and expected value. Probability
distributions will be calculated from both theoretical and empirical probabilities and will be extended to find expected value. To
culminate the unit, students will examine using expected value to make decisions about insurance, participating in a game, etc.
STANDARDS
S.MD.1 (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the
corresponding probability distribution using the same graphical displays as for data distributions.
S.MD.2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
S.MD.3 (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be
calculated; find the expected value. For example: find the theoretical probability distribution for the number of correct answers obtained
by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under
various grading schemes.
S.MD.4 (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned
empirically; find the expected value. For example: find a current data distribution on the number of TV sets per household in the United
States, and calculate the expected number of sets per household. How many TV sets would you expect in 100 randomly selected
households?
S.MD.5a Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. Find the
expected payoff for a game of chance. For example: find the expected winnings from a state lottery ticket or a game at a fast-food
restaurant.
S.MD.5b Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. Evaluate
and compare strategies on the basis of expected values. For example: compare a high-deductible versus low-deductible automobile
insurance policy using various, but reasonable, chances of having a minor or major accident.
MATH PRACTICES
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning
LITERACY STANDARDS
L.1
L.2
L.3
L.4
L.6
L.7
L.8
L.9
Learn to read mathematical text - - problems and explanations
Communicate using correct mathematical terminology
Read, discuss, and apply the mathematics found in literature, including looking at the author’s purpose
Listen to and critique peer explanations of reasoning
Represent and interpret data with and without technology
Research mathematics topics or related problems
Read appropriate text, providing explanations for mathematics concepts, reasoning or procedures
Apply details of mathematical reading / use information found in texts to support reasoning, and develop a “works cited document”
for research done to solve a problem.
7/02/2013 YCS PRE-CALC UNIT 9: PROBABILITY, RANDOM VARIABLES AND EXPECTED VALUE 2013-2014 1
TEACHER NOTES
MOTIVATION
1. Ask students the following questions:
a) Would you like to be able to calculate expected winnings from the lottery?
b) Would you like to be able to calculate the odds of winning a 50-50 raffle?
c) Would you like to be able to make informed decisions when purchasing car insurance, life
insurance, and house insurance?
d) Would you like to be able to calculate an expected grade if you guessed on all the questions of a
multiple choice test?
e) Would you like to be able to calculate the pay-off of a game at an internet café?
Tell students that by the time they complete this unit, they will be able to calculate these values.
2. Preview expectations for the end of the Unit
3. Have students set both personal and academic goals for this Unit
TEACHER NOTES
TEACHING-LEARNING
Note to teacher: sections in the textbook that may be of help are 13-1, 13-3, 13-6, 14-1, 14-2
Vocabulary:
Sample space
Theoretical probabilities
Tree diagram
Combinations
Empirical probability
Random variable
Expected value
Permutations
Probability distribution
Population
Discrete random Variable
Continuous random Variable
Binomial theorem
1. Review probability, tree diagrams, permutations, combinations and the binomial theorem used in
probability. One method of reviewing this material is to divide students into five groups and give each
group a concept. They are to present a definition to the class, present a formula if there is one, show
how the formula was derived, and present 2 examples. They are also to have 5 additional problems with
answers on paper for use as homework problems. (S.MD.3, MP.2, MP.4, MP.5, MP.7, L.1, L.2, L.4, L.7,
L.8)
After completing this activity, to familiarize students with concepts of probability, the activities presented
in the following link are recommended: http://illuminations.nctm.org/LessonDetail.aspx?id=L290
2. Discuss random variable which is like a function in that it assigns a number to an event. Examples of
random variables are
 X=
 X = Exact amount of rain in inches tomorrow
1″ to 2″ infinite number of possibilities
 X = number facing up on a throw of a fair dice
The first and third examples are examples of discrete random variables, there are a countable number
of outcomes. The second example is an example of a continuous random variable, infinite number of
outcomes. There are infinitely many numbers between 1 and 2. Once students understand the concept
of a random variable, have them define a discrete random variable and a continuous random variable,
present all the examples and have the class choose one of each as the best examples. A small prize
such as extra points, candy, etc could be given to the winners. (S.MD.1, MP.1, MP.2, MP.3, MP.4,
MP.7, L.2, L.4, L.6)
https://www.khanacademy.org/math/probability/random-variables
topic/random_variables_prob_dist/v/random-variables video explaining random variables
3. There are two types of probability, theoretical probability and empirical probability. Have students
7/02/2013 YCS PRE-CALC UNIT 9: PROBABILITY, RANDOM VARIABLES AND EXPECTED VALUE 2013-2014 2
TEACHER NOTES
TEACHING-LEARNING
research the two terms, find workable definitions and examples of each.
Reference page 877 in the textbook. (S.MD.3,S.MD.4, MP.1, MP.2, MP.4, MP.7, L.1, L.2, L.7, L.8)
http://www.regentsprep.org/Regents/math/ALGEBRA/APR5/theoProp.htm video explaining empirical
and theoretical probability.
4. To discuss probability distributions, examine an empirical probability problem where a random
sample was taken of 200 households to determine the number of children per household. The results
are listed in the table below where the random variable X =
:
# of children
0
1
2
3
4
5
6
7
8
frequency
50
35
70
27
7
2
4
0
5
probability
Have students use this information to make a probability distribution graph by finding the probability
for each frequency (refer to textbook page 901). Note the sum of the probabilities is one. Reinforce
making probability distribution graphs with several examples of data. (S.MD.1, MP.2, MP.4, MP.8, L.2,
L.6)
http://www.zweigmedia.com/ThirdEdSite/tutstats/frames8_1.html tutorial with examples to work out and
answers given, random variables, discrete and continuous, empirical probability, probability distributions
TEACHER GENERATED ASSESSMENT
5. Use the example in T/L #4, to calculate the expected value of the random variable X. Prior to
calculating the expected value, ask students to make an educated guess as to what they might think the
expected value would be. In data sets, students calculated the mean (arithmetic average) as the
measure of center for the data. For probability distributions, the expected value (weighted average) is
the measure of center. To calculate the expected value (preferably on a spread sheet) multiply the
values by the probabilities from the table in T/L#4 and add them.
0*0.250 + 1*0.175 + 2*0.350 + 3*0.135 + 4*0.035 + 5*0.010 + 6*0.02 + 7*0 + 8*0.25 = 1.79
This means on average there are 1.79 children per household. How would the number change if the
households with no children were left out? Reinforce with more examples. (S.MD.2, S.MD.4, MP.1,
MP.2, MP.4, MP.5, MP.6, MP.7, L.2) the following website explain how to find expected value
http://www.youtube.com/watch?v=51fIfbLMP60&list=PL47439F7E5E0B9D48&index=14
6. Students have just found an expected value for an empirical probability distribution. Now they will work
with a theoretical probability model and find the expected value. Example: find the theoretical
probability distribution for the number of correct answers obtained by guessing on all five questions of a
multiple-choice test where each question has four choices and find the expected grade. This is a
binomial probability distribution where : n = 5, p = ¼ , q = ¾
The probability distribution is:

P(x=0) = 5C0


P(x=1) = 5C1
P(x=2)
= 5C2

P(x=3) = 5C3

P(x=4) = 5C4
7/02/2013 YCS PRE-CALC UNIT 9: PROBABILITY, RANDOM VARIABLES AND EXPECTED VALUE 2013-2014 3
TEACHING-LEARNING

TEACHER NOTES
P(x=5) = 5C5
The expected value is:
Another example is: In a game at a carnival players roll a 10 sided die. If they roll a number less than 8
they make $8, if they roll anything else, they lose $6. What are the expected winnings or loss?
The probability distribution is
The expected winnings is
(S.MD.2, S.MD.3, S.MD.5a, MP1, MP.2, MP.4, MP.5, MP.6, MP.8, L.1, L.2, L.6)
http://hotmath.com/hotmath_help/topics/probability-distribution.html
7. Expected values can be used to make decisions about purchasing car insurance. Discuss the following
example: Your Mom has just purchased a new Toyota Tundra and has asked you for help in
determining whether or not to buy car insurance. The insurance costs $1250 a year with a $150
deductible. The probability of your Mom having an accident in the next year is 8% and the average cost
of repairs in an accident for the Tundra is $4500.

What is the expected value of purchasing the insurance for the year?
Ans: 1250 +150*0.08 = $1262

What is the expected value if she does not buy the insurance?
Ans: 0.08*4500 = $360
When examining this problem at face value, it seems like it would be a good idea not to buy the
insurance. Discuss with students the hidden expenses that are not considered here such as the fines
and court costs associated with driving without insurance, the probability that she had an accident and
the cost of repairs exceeded $4500, etc. Reinforce with additional problems using expected value to
make decisions. (S.MD.2, S.MD.3, S.MD.5a&b, MP.1, MP.2, MP.4, MP.5, MP.6, MP.8, L.2, L.6)
Some links that may be of help are: http://www.freemathhelp.com/forum/threads/55235-LotteryWinnings-calculate-expected-net-winnings easily shows how to calculate expected winnings
http://www.ehow.com/how_11383670_calculate-expected-value-decision-trees.html expected value of
an investment
http://www.ehow.com/how_6513954_calculate-expected-values.html expected winnings from lottery
http://www.rocklin.k12.ca.us/staff/cmcnabbnelson/cpm.geo/unit%20Probability/expectedvalue.pdf
overview of expected value – lottery, casino games, and stocks
http://www.thepokerbank.com/strategy/mathematics/expected-value/ use expected value on coin flip
and poker to make decisions
http://brainmass.com/business/finance/243108 use expected value to make decision about a company
http://www.rocklin.k12.ca.us/staff/cmcnabbnelson/cpm.geo/unit%20Probability/expectedvalue.pdf
decision to buy stocks
http://answers.yahoo.com/question/index?qid=20071126200506AAcbtrc use expected value for
insurance
7/02/2013 YCS PRE-CALC UNIT 9: PROBABILITY, RANDOM VARIABLES AND EXPECTED VALUE 2013-2014 4
TRADITIONAL ASSESSMENT
TEACHER NOTES
1. Paper-pencil test with M-C questions.
TEACHER CLASSROOM ASSESSMENT
1. 2 and 4 point questions
2. Other projects, worksheets, etc.
TEACHER NOTES
AUTHENTIC ASSESSMENT
1. Have students evaluate goals for the unit.
TEACHER NOTES
2. Students are to find empirical data showing a value with its frequency, dealing with a topic of
their interest. Using these data, they are to discuss two strategies where expected value may be
used to determine which would be the better of the two strategies. They are to show two
probability distributions, calculate the expected values, and state which the better choice is. For
example: using statistics about two baseball pitchers, compute the expected values of each and
then using that information to decide which pitcher would be better to use.
3. Students are to research purchasing car insurance, write a 500 word report and include a “works cited”
page. They need to include laws for insurance in the state of Ohio, what deductibles are and how they
affect the premiums, items that decrease the amount of the premiums, items that increase the amount of
the premiums, maximum number of moving violations that will increase the cost of the premium, and an
insurance quote using either their car or their parent’s car. (S.MD.5b, MP.1, MP.4, L.2, L.3, L.7, L.9)
#2 AUTHENTIC ASSESSMENT RUBRIC
ELEMENTS OF THE
PROJECT
2 empirical data sets with a
value and frequency
Probability distributions
Calculate expected values
State decision as to which is
the better choice and
reasoning for the decision
0
Did not find data
sets
Did not calculate
either probability
distribution
Did not calculate
either expected value
Did not state a
decision
1
Specified one data set
Calculated one probability
distribution correctly
Calculated one expected
value correctly.
Stated a decision without
reasoning
2
Specified two data sets that are
empirical
Calculated both probability
distributions correctly
Calculated both expected values
correctly
Stated a decision with reasoning
7/02/2013 YCS PRE-CALC UNIT 9: PROBABILITY, RANDOM VARIABLES AND EXPECTED VALUE 2013-2014 5