Download t/l/#6 titanic problem - Youngstown City Schools

YOUNGSTOWN CITY SCHOOLS
MATH: GEOMETRY
UNIT 6: APPLICATIONS OF PROBABILITY (5 WEEKS)
Synopsis: Students will rely on the probability concepts they learned in the lower grades and use those to expand their
knowledge. Using two-way tables, they will explore mutually exclusive events, independent events, conditional probability, and
the probability of inclusive events. The will become familiar with rules of probability to compute probability of compound events
and use probability to evaluate outcomes.
STANDARDS
S.CP.1, 2,3,4,5
Build on work with two-way tables from Algebra I Unit 3 (S.ID.5) to develop understanding of conditional probability and independence.
S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as
unions, intersections, or complements of other events (“or,” “and,” “not”).
S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their
probabilities, and use this characterization to determine if they are independent.
S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the
conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the
probability of B.
S.CP.4, 5
Build on work with two-way tables from Algebra I Unit 3 (S.ID.5) to develop understanding of conditional probability and independence.
S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified.
Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example,
collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the
probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for
other subjects and compare the results.
S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer
S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in
terms of the model.
S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the
answer in terms of the model.
S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
S.MD.6, 7
This unit sets the stage for work in Algebra II, where the ideas of statistical inference are introduced. Evaluating the risks associated with
conclusions drawn from sample data (i.e. incomplete information) requires an understanding of probability concepts.
*Use S.MD.6,7 as an Enrichment Support; Excluded from CFAs; See Model Curriculum
S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
S.MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at
the end of a game).
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YCS Geometry Unit 6 Applications of Probability 2013-2014 1
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 Learn to read mathematical text (including textbooks, articles, problems, problem explanations)
L.2 Communicate using correct mathematical terminology
L.4 Listen to and critique peer explanations of reasoning
L.5 Justify orally and in writing mathematical reasoning
L.6 Represent and interpret data with an without technology
L.7 Research mathematics topics or related problems
MOTIVATION
TEACHER NOTES
1. Illuminations illustrates probability by engaging in an activity involving a forest fire. To motivate students
to study probability, have them work on the activity.
http://illuminations.nctm.org/ActivityDetail.aspx?ID=143
2. Preview expectations for the end of the Unit or the grading period
3. Have students set both personal and academic goals for this Unit.
TEACHING-LEARNING
TEACHER NOTES
http://illuminations.nctm.org/ActivityDetail.aspx?ID=67
http://www.cut-the-knot.org/Probability/Dictionary.shtml link that discusses all the topics in this unit
http://www.teachengineering.org/view_activity.php?url=collection/drx_/activities/drx_probability/drx_probabili
ty_activity3.xml discusses card games in relation to probability, permutations and combinations,
encourages students to make their own games.
http://www.jamesrahn.com/Geometry/activities/geometric%20probability%20activiities.pdf additional
problems on
Vocabulary:
Sample space
Probability
Mutually exclusive
Independent events
Complement
Inclusive
Conditional probability Union
Intersection
Outcome
With replacement
Without replacement
Multiplication rule
Addition rule Permutations
Combinations
1. Review the basics of probability: (S.CP.1, MP.1, MP.2, MP.4, MP.5, MP.6, MP.7, L.1, L.2, L.6)
a) Given a segment AB which has length 14 cm. A point C is on AB such that AC is 5 cm. Q is
somewhere on AB. Find the probability that Q is on AC? BC? Not on AC?
b) Given a circle W with A and Y being endpoints of a diameter. Point Z is some other point on circle
W. What is the probability that <XZY is a right angle?
c) A rectangular garden is 150 ft. x 125 ft. and has a flower bed that is 30 ft. on each side. Find the
probability that a bee in the garden is somewhere in the flower bed.
d) A box contains four rectangles, 3 trapezoids, and 6 triangles, what is the probability that a
rectangle will be chosen and then a triangle with replacement? Without replacement?
e) There are 10 angles either in or on a circle. If we would like group them three at a time, how many
possibilities are there?
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YCS Geometry Unit 6 Applications of Probability 2013-2014 2
TEACHING-LEARNING
TEACHER NOTES
2. When describing events, refer to them as subsets of a sample space (set of outcomes) using
characteristics of the outcomes, or as unions, intersections, or complements or other events. For
instance in 1-a, segment AB is the sample space and the subsets are AC and CB. The complement of
AC is BC. (S.CP.1, MP.4, L.2)
3. Discuss mutually exclusive events and independent events and their differences.
http://cnx.org/content/m16837/latest/
(S.CP.1, S.CP.2, MP.1, MP.2, MP.4, MP.5, MP.6, MP.7, L.1, L.2, L.6)
 Mutually exclusive events occur if event A happens, then event B cannot, or vice-versa. The two
events "a quadrilateral is a parallelogram" and "a quadrilateral is a trapezoid" are mutually
exclusive events. When calculating the probabilities for exclusive events you add the probabilities.
For example a bag contains 12 quadrilaterals, 4 of which are parallelograms, 6 of which are
trapezoids and two are not special quadrilaterals. What is the probability that you would pick out a
parallelogram or a trapezoid? Have students work on this before explaining: 4/12 + 6/12 =
8/12=2/3=66.7%. In mutually exclusive events P(A and B) = 0, you are finding P(A or B) =
P(A)+P(B) = P(A
B) Illustrate this with a Venn diagram. An interesting view on applying Venn
diagrams to probability can be found on the following link:
http://www.amstat.org/publications/jse/v6n1/warner.html
 Independent events are ones in which the outcome of event A, has no effect on the outcome of
event B. Such as "RS is 13 in. and T is on RS and is 5 inches from R" and "WZ is 25 inches and X
is on WZ and is 12 inches from Z". When calculating the probabilities for independent events you
multiply the probabilities. You are effectively saying “what is the chance of both events happening
bearing in mind that the two were unrelated”. Using the stated example, what is the probability that
point Q will lie on RT and that point Y will lie on XZ? Have students work on this before explaining:
P(A) *P(B) = 5/13*12/25=12/65=18.5%).
 Have students break into groups and discuss the following problem: What happens if we want to
throw 1 and 6 with two rolls of a die in any order? Examine the first roll, decide if it is mutually
exclusive or independent, and then find its probability. Examine the second roll, decide if the
second roll is mutually exclusive or independent of the first roll, then find the probability.
(Ans: In the first roll, the probability of throwing 1 or 6 with the first die are mutually exclusive, so
the probability is the sum of the two probabilities, 1/6 + 1/6 = 1/3.
In the second roll, the probability of the second die being favorable is still 1/6 as the second die can
only be one specific number, a 6 if the first die is 1, and vice versa. The second roll is independent
of the first roll. Therefore the probability of throwing 1 and 6 in any order with two dice is 1/3 x 1/6
= 1/18. Note that we multiplied the last two probabilities.)
District Generated Assessment
4. Conditional Probability: Below are three links that may be of help in teaching and having students
discover conditional probability. The first link discusses conditional probability using the multiplication
rule for probability P(A/B) =
and explains how it is derived from P(A+B) = P(A) * P(B/A). The
second link uses the TI NSpire and is an excellent activity to introduce conditional probability. The third
link gives a good definition of conditional probability and uses a two-way table in its explanation.
(S.PC.1, S.PC.2, S.CP.3, S.CP.4, S.CP.6, S.CP.7, S.CP.8, MP.1, MP.2, MP.4, MP.5, MP.6, MP.7, L.2,
L.6)
http://www.mathgoodies.com/lessons/vol6/conditional.html
http://education.ti.com/en/us/activity/detail?id=BC6AFC94F54A4AAB9C5284A64824A950
http://algebralab.org/lessons/lesson.aspx?file=Algebra_ConditionalProbability.xml
 To introduce conditional probability, look at the problem: there are five angles whose measures
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YCS Geometry Unit 6 Applications of Probability 2013-2014 3
TEACHING-LEARNING
500,
600,
900,
TEACHER NOTES
1200
400.
are
and
Suppose you choose two angles without replacement. Given that
the first angle is acute, what is the probability the second angle is obtuse? Using the formula for
conditional probability where A is the probability that an acute angle is chosen and B is the
probability an obtuse angle is chosen: P(A/B) =
with more examples.
= 25%. Reinforce
 Have students create a two-way table with the following information: in a bag there are 10 blue
trapezoids,15 red trapezoids, 8 blue rectangles and 12 red rectangles. The table may look like:
Blue
Red
total
Trapezoids
10
15
25
Rectangles
8
12
20
Total
18
27
45
 Review the terms sample space as being the polygons in the bag and outcome which would be
perhaps “What is the probability that a trapezoid is chosen?” Question students or have students
create problems which illustrate mutually exclusive events and independent events using the table.
 Use the information in the table to discuss the formula for conditional probability. For example,
what is the probability that a rectangle with be chosen, given it is red? Stated differently, what is
the probability of choosing a red rectangle? Ask students several of these questions to reinforce
conditional probability.
5. Have students research the term inclusive events as defined in probability. Discuss their definitions,
have students critique them and then arrive at a class definition. Once students understand the
concept, use the addition rule for probability to work problems.
P(A or B) = P(A) + P(B) − P(A+B)
Suggested example 1: A game board is made from a rectangle that is 10 ft X12 ft. and a circle
attached to the rectangle whose diameter is one third the longer side of the rectangle as shown
below. What is the probability a dart will lie on the square or the circle? Using the above rule, the
solution would be: probability of landing on the square + probability of landing on the circle –
probability of landing on half the circle.
Suggested example 2: What is the probability of randomly choosing a point inside the cylinder but not
inside the cone that has the same base and height as the cylinder and is inscribed in the cylinder?
Ans: Probability of landing in the cylinder − probability of landing in the cone.
http://stattrek.com/probability/probability-rules.aspx states rules of probability
(S.CP.7, MP.1, MP.2, MP.3, MP.4, MP.5, MP.6, MP.7, L-1, L-2, L-4, L-6, L-7, L-8)
6. Having discussed conditional probability and independence, students need to engage in an activity
where independence is defined by conditional probability. If A and B are independent then P(A/B) =
P(A) and P(B/A) = P(B). Also P(A∩B) = P(A) P(B). An example is if a coin is flipped twice, the two
events are independent of each other. There is a misconception that the independency or dependency
of two events relates to their having or not having an empty intersection. The case of A and its
complement A is an example of two dependent events with empty intersection. As reinforcement of
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YCS Geometry Unit 6 Applications of Probability 2013-2014 4
TEACHING-LEARNING
TEACHER NOTES
conditional probability, engage students in the Titanic problem attached on page 8.
Additional examples can be found on http://www.slideshare.net/leingang/worksheet-conditionalprobability-solutions (S.CP.2, S.CP.3, MP.1, MP.2, MP.4, L.1, L.2, L.6)
7. Have students find or create problems containing the concepts of conditional probability and
independence in everyday situations and share them with the class. (S.CP.2, S.CP.3, S.CP.5, MP.1,
MP.2, MP.4, MP.7, L.1, L.2, L.4, L.6, L.7)
8. Discuss finding probabilities of compound events using permutations or combinations.
 You are responsible for purchasing pizzas for your group. The store has 15 toppings for pizza, 8 of
which are meat and 7 are nonmeat. You state that you would like each pizza to have 5 toppings.
What is the probability that you would get 3 meat and 2 nonmeat toppings on a pizza? Ans:
C(8,5)*C(7,5)/ C(15,5)
 Use the above example and have students create their own problems probabilities of compound
events using permutations or combinations
(S.PC.9, MP.1, MP.2, MP.4, MP.5, MP.7, L.2, L.6)
TEACHER GENERATED ASSESSMENT
9. When playing games or evaluating results from product testing, probability is used to make fair
assessments or decisions. Look at a spinner such as the one below: (S.CP.4, S.CP.6, S.MD.6, S.MD.7,
MP.1, MP.2, MP.3, MP.4, MP.5, MP.6, MP.7, L.2, L.4, L.5, L.6)
 Using a protractor, determine the probabilities of the arrow landing on each of the sections. Would
you like to play a game using this particular spinner? If so explain the rules of the game so this
spinner would be advantageous to you. If not, explain why not. Do you believe this to be a fair
spinner?
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YCS Geometry Unit 6 Applications of Probability 2013-2014 5
TEACHING-LEARNING
TEACHER NOTES
 Engage students with the bulls-eye. Have them copy it on a piece of paper and find the
probabilities of a dart landing on each of the colors, assign points to each area and present to the
class giving an explanation for the assignment of points. Have the class critique their point
assignments.
https://ccgps.org/S-MD_V6RM.html
Below is a chart showing the results from a survey of 500 students on the cost of an I-phone
Girls
Guys
Too expensive
0.50
0.20
Cost is appropriate
0.12
0.28
Not priced high enough
0.03
0.12
 What is the probability that a guy thinks the I-phone is too expensive? Ans:
= .20/.60 = 33%
 What is the probability that a girl thinks the cost is appropriate? Ans:
= .12/.65 = 18.5%
 Ask students the question “How can Apple use this data to help them sell I-phones?”
Included on page 9 are problems dealing with geometric probability.
TRADITIONAL ASSESSMENT
TEACHER NOTES
1. Paper-pencil test with M-C questions
TEACHER CLASSROOM ASSESSMENT
TEACHER NOTES
1. quizzes
2. other assessments
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YCS Geometry Unit 6 Applications of Probability 2013-2014 6
AUTHENTIC ASSESSMENT
1. Students evaluate goals set at the beginning of the unit or on a weekly basis.
TEACHER NOTES
2. Create a game that uses at least 5 geometric figures for the game board. Students may not use
a bull’s eye or spinner.
a) Divide the board into sections and find the probability of each section.
b) State rules for the game
c) State directions as to how to play the game
d) Determine if the game is fair and state reasons for your decisions
(S.MD.6, S.MD.7, MP.2, MP.4, MP.5, MP.6, L.2)
Elements of the Project
Create game
Found probability of each
section
State rules of the game
State directions
Determine fairness of
game and reasons
0
Did not
attempt
Did not
attempt
Did not
attempt
Did not
attempt
Did not
attempt
RUBRIC
1
Used one geometric
figure
Found probability of
one section
NA
2
Used three geometric
figures
Found probability of
three sections
NA
NA
NA
Stated fairness only
NA
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3
Used at least five
geometric figures
Found probability of at
least five sections
Stated rules of the
game
Directions stated
Stated fairness and
gave reason
YCS Geometry Unit 6 Applications of Probability 2013-2014 7
T/L/#6 TITANIC PROBLEM
On the morning of April 10, 1912 the Titanic set sail from Southampton, England on its maiden voyage to New
York City. On April 14, just four days into the trip, the Titanic struck an iceberg and in just over 2 hours the
"unsinkable" Titanic was gone with a loss of 1,490 lives. The Titanic had been outfitted with 20 lifeboats,
enough to hold about 54% of the passengers.
The table below gives a count of the survivors and victims of the Titanic sinking. The data are classified by type
of passenger (first class, second class, third class, crew member) and survival status. The data for this analysis
were compiled from the initial casualty estimates, as reported in the Official Board of Trade Inquiry Report,
written originally in 1912. NOTE: For a complete list of passenger names see the poster on my office door. No,
Leonardo DiCaprio is NOT listed.
survived died
first class 203 122
second class 118 167
third class 178 528
crew class 212 673
Question 1. What is the probability that a randomly selected passenger survived?
Question 2. What is the probability that a passenger survived given that the passenger was in second class?
Question 3. What is the probability that a passenger survived and was in second class?
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YCS Geometry Unit 6 Applications of Probability 2013-2014 8
Geometric Probability
Examples
The Parachute Problem
A parachutist jumps from an airplane and lands in a square field that is 2 kilometers on each side. In each corner of the field
there is a large tree. The parachutist’s ropes will get tangled in the tree if she lands within 1/11 kilometer of its trunk. What is
the probability that she will land in the field without getting caught in a tree?
The Fairground Problem
At a state fair a game is played by tossing a coin onto a large table ruled into congruent squares. If the coin lands entirely
within some square and doesn’t touch any edges, the player wins a prize. If the coin touches or crosses the edge of any
square, the player loses. Suppose the squares are S units on a side and the coin has a radius of R units. What is the
probability of winning this game?
The Triangle Problem
The students in a geometry class are considering when three line segments will form a triangle. They know that a triangle
will be formed if and only if the sum of the lengths of any two segments is greater than the length of the third segment. The
class has two wooden dowels, one of which is twice as long as the other. The students break the longest dowel at a random
spot, so that they now have three dowels. What is the probability that the three lengths form a triangle?
The Tape Recorder Problem
A tape recording is made of a meeting between two businessmen. Their conversation starts at the 21st minute on the tape
and lasts only 8 minutes. The tape will record for 60 minutes. While playing back the tape one of the businessmen
accidentally erases 15 minutes of the tape, but he doesn’t know where.
1. Find the probability that the entire conversation was erased.
2. Find the probability that some part of the conversation was erased.
3. Suppose the exact position of the conversation on the tape is not known, except that it began sometime after the 21st
minute. Find the probability that the entire conversation was erased.
Above problems from: Geometric Probability. New topics for secondary school mathematics. NC School of Science and
Mathematics. National Council of Teachers of Mathematics, Inc. 1988.
Answers on
http://courses.ncssm.edu/math/tcm/TCM2011/talks/gann/Geometric%20Probability%20solutions.pdf
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YCS Geometry Unit 6 Applications of Probability 2013-2014 9