Download Geometry Unit 5 Formative Itemsx

Geometry Items to Support Formative Assessment
Unit 5: Probability of Compound Events
Part I: Probability of Compound Events
Understand independence and conditional probability and use them to interpret data.
S.CP.A.1 Describe events as subsets of 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.A.1 Item 1:
1. Far Flight High School has 162 students enrolled in French, 294 students enrolled in Spanish,
and 78 students taking both languages. The remaining 132 students do not take a language. Find
each of the probabilities below and use appropriate notation to summarize the event.
a.
b.
c.
What is the probability that a student takes French or Spanish or both?
What is the probability that a student takes French and Spanish?
What is the probability that a student does not take French?
2. How would you describe the students you would include if you were considering the
complement of P(French Ç Spanish)? What is the probability of this occurring?
Solution:
1.
a.
P(French È Spanish) » 0.74
b.
P(French Ç Spanish) » 0.15
c.
P(Frenchc) » 0.68
2.
The students who are not taking a language would be the complement French
The probability is approximately 0.26.
Spanish.
S.CP.A.1 Item 2:
A local coffee shop, Fourbucks, is interested in collecting statistics about its patrons to determine
next steps for developing future marketing strategies. The Venn diagram below represents the
results of the survey conducted about what patrons purchased during their visit.
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Use the information from the Venn diagram to write and answer three questions using the correct
notation. Be sure to include at least one intersection, union, and complement.
Possible Solutions:
If a patron is selected at random, determine the following:
a. P(purchasing a drink itemc) =
25
= 0.25
100
95
= 0.95
100
15
c. P(Purchasing a Drink Item Ç Purchasing a Food Item) =
= 0.15
100
b. P(Purchasing a Drink Item È Purchasing a Food Item) =
S.CP.A.1 Item 3:
Write a scenario based on the survey results shown below. Then write 3 probability problems
based on your scenario.
Possible Solutions:
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In a sample of 500 randomly selected people, 187 said they owned an mp3 player; 285 said they
owned a smartphone; 75 reported they owned both, and 103 said they didn’t own either.
P(smartphone È mp3 player) = 0.794
P(owns an mp3 playerc) = 0.626
P(smartphone) = 0.57
S.CP.A.1 Item 4:
250 people in the lodge at Moose Mountain Ski Resort were asked if they were there to ski or
snowboard. The results of the survey are shown in the two-way table below.
Skis
Doesn’t Ski
Total
Snowboards
95
58
153
Doesn’t Snowboard
60
37
97
Total
155
95
250
If a person at the lodge is selected at random, determine the probability of the events below.
1.
P(skis)
2.
P(doesn’t participate in either sport)
3.
P(snowboards È skis)
Solution:
1.
P(skis) = 0.24
2.
P(doesn’t participate in either sport) = 0.148
3.
P(snowboards È skis) = 0.852
S.CP.A.1 Item 5: (Adapted from NCTM Mathematics Assessment Sampler Grades 9-12)
Twenty-five people watched a movie showing at the AMC Theater. Of them, fifteen order a
drink, ten order popcorn, and nine order candy. Two people order all three items, one orders a
drink and candy, three order a drink and popcorn, and one orders popcorn and candy.
a. Display the results in a Venn diagram.
b. What is the probability that a moviegoer who buys a drink will also buy popcorn.
Solution:
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a.
b. 5/15 or 1/3
Use the rules of probability to compute probabilities of compound events in a uniform
probability model.
S.CP.B.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.B.7 Task:
The Bar Harbor Boating Company in Maine offers 3 boating tours: Just Whale Sightings, Just
Puffin Sightings, and Whale and Puffin Sightings. The manager would like to guarantee the
sighting of whales and/or puffins on all of the tours or he would give his customers a full refund,
however, he is worried he might lose too much money. If the proportion of sighting only a
whale is 0.75, the proportion of sighting only a puffin is 0.95, and the proportion of sighting a
whale and puffin is 0.72, set up a two-way table. Then, use what you know about calculating
probability to make a recommendation to the manager whether or not he should guarantee
sightings for each tour. Include a two-way table to support your recommendation.
Sample Answer:
Puffins
No Puffins
Total:
Whale
0.72
0.03
0.75
No Whales
0.23
0.02
0.25
Total:
0.95
0.05
1.00
P(Puffin È Whale) = 0.95+0.75-0.72= 0.98
P(Puffin Ç Whale) = 0.72
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I would recommend that the manager guarantee the sighting of puffins on the Just a Puffin tour,
since the probability of seeing a puffin on a random day is 0.95. I would not recommend the
manager offer the guarantee for the sighting of whales on Just the Whale tour, since the
probability of seeing a whale on a random day is only 0.75. For the Whale and Puffin tour, I
would recommend that the manager offer a guarantee for the sighting of either puffins or whales,
since the probability of sighting both a puffin and whale on a random day is only 0.72, but the
probability of sighting a puffin or a whale is 0.98.
S.CP.B.7 Item 1:
At Bayside High School 42% of their students participate in a sport after school, 27% of their
students participate in an after school club, and 15% of students participate in both a club and
sport after school. The principal, Mr. Belding, would like to determine how involved their
students are in after school activities. Create a two-way table to display the data, and find the
probability that a randomly selected Bayside student will participate in either a club or sport after
school.
Answer:
Participates in a
Sport
Does Not Participate
in a Sport
Total
Participates in a
Club
0.15
0.12
0.27
Does not participate
in a Club
0.27
0.46
0.73
Total
0.42
0.58
1.00
P(Club È Sport): 0.42+0.27-0.15= 0.54
S.CP.B.7 Item 2:
Below is the two-way table representing a survey of the proportion of 100 random moviegoers
that were interviewed about whether or not they saw the first two movies in The Hunger Games
series.
Saw The Hunger
Games
Did not see The
Hunger Games
Total
0.29
0.04
0.33
Did not see Catching 0.13
0.44
0.67
Saw Catching Fire
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Fire
Total
0.42
0.58
1.00
The CFO of the local movie theatre conducted the interview with his patrons to determine
whether or not to show the third movie. To help with his decision, he determined the probability
that a randomly selected patron had seen either The Hunger Games or Catching Fire. Below is
his work:
P(The Hunger Games È Catching Fire) = 0.42 + 0.33 = 0.75
Determine if he is correct and explain why or why not. If he is incorrect, provide the correct
probability.
Possible Solution:
He is incorrect in his calculations, since he forgot to subtract the patrons that saw both
The Hunger Games and Catching Fire. If he does not subtract these patrons, then they will count
twice. The correct probability is P(The Hunger Games È Catching Fire) = 0.42+0.33–0.29=0.46.
S.CP.B.7 Item 3: (From NCTM Mathematics Assessment Sampler Grades 9-12)
A warning system installation consists of two independent alarms having probabilities of
operating in an emergency of 0.95 and 0.90, respectively. Find the probability that at least one
alarm operates in an emergency. Show your work.
a.
b.
c.
d.
e.
0.995
0.975
0.95
0.90
0.855
(Note: This problem was originally a TIMMS item. It allows students to solve the problem
using a variety of approaches, some more complicated than others. The phrase “at least” gives
students the opportunity to show their understanding of the complement of the event that neither
of the alarms works.)
Solution: a
Strategy 1: (1 - (0.05)(0.1)) = 1 - 0.005 = 0.995
Other valid student work:
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
S.CP.B.7 Item 4: (Note: This table is also used with an item for S.CP.A.3)
The table below represents the distribution of students that attended the Spirit Dance.
Freshmen
Sophomores
Juniors
Seniors
Attended Spirit Dance
240
185
265
270
Did Not Attend Spirit Dance
60
105
45
30
1. Find the probability that a student selected at random is a junior OR is a student that
attended the spirit dance.
2. Find the probability that a student selected at random is a senior OR is a student that did
not attend the spirit dance.
Solutions:
1.
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960 + 45 1005
=
1200
1200
Or
P(A or B) = P(A) + P(B) – P(A and B)
310 960
265 1005
+
=
1200 1200 1200 1200
2.
240 + 270 510
=
1200
1200
Or
P(A or B) = P(A) + P(B) – P(A and B)
300 240
30
510
+
=
1200 1200 1200 1200
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.