Geometry Items to Support Formative Assessment
Unit 5: Probability of Compound Events
Part II: Conditional Probability
Understand independence and conditional probability and use them to interpret data.
S.CP.A.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.A.3 Task:
The following cards have been placed in a bag:
1. Define Events A and B in a way that makes the events independent. Verify your
conclusions.
This task has a number of possible solutions. A few are shown below:
Sample Solution A:
Event A: “You pick a face card.”
Event B: “You pick a red card.”
Since there are 4 face cards in the bag, P(A) =
face cards, so P(A|B) =
1
. Of the 6 red cards, 2 of the cards are
3
1
. Since P(A) = P(A|B), the two events are independent.
3
Sample Solution B:
Event A: “You pick a card with an even number.”
Event B: “You pick a red card.”
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Since there are 4 even cards in the bag, P(A) =
even number cards, so P(A|B) =
1
. Of the 6 red cards, 2 of the cards are
3
1
. Since P(A) = P(A|B), the two events are independent.
3
2. Define Events A and B so that the two events are not independent. Verify your
conclusions.
Sample Solution:
Event A: “You pick a number card with value less than 10.”
Event B: “You pick a red card.”
7
.
12
Of the 6 red cards, 3 of the cards are number cards with value less than 10, so P(A|B) =
1
. Since P(A) ¹ P(A|B), the two events are not independent.
2
Since there are 7 number cards with value less than 10 in the bag, P(A) =
S.CP.A.3 Item 1:
There are 10 lockers outside Ms. McBee’s math class. The locker numbers are 311-320. Ms.
McBee has placed an iTunes gift card in six of the lockers. She has written the locker numbers
on cards and placed the cards in a bag. A student will draw a locker number at random to see if
he/she has won a prize.
Event A: “You pick a locker with an iTunes gift card.”
Event B: “You pick a locker number that is an even number.”
1. If Ms. McBee placed two of the gift cards in a locker with an even-numbered locker
number and the remaining four gift cards in lockers with odd-numbered locker numbers,
then determine if events A and B are independent events. Explain your reasoning.
Possible Solutions:
Since there are 10 total lockers and 6 that have gift cards, P(A) =
3
5
2
5
Knowing that the locker selected will be even does impact the probability, so the events
are not independent.
Out of the 5 even-numbered lockers, only two have gift cards, so P(A|B) =
Alternate Solution:
3
P(A) =
5
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
P(B) =
1
2
2
10
3
P(A) × P(B) =
10
Since P(A) × P(B) ¹ P(A and B), then the two events are not independent.
P(A and B) =
2. If Ms. McBee placed three of the gift cards in a locker with an even-numbered locker and
the remaining three gift cards in lockers with odd-numbered locker numbers, determine if
events A and B are independent events. Explain your reasoning.
Sample Solutions:
Since there are 10 total lockers and 6 that have gift cards, P(A) =
3
5
3
5
Knowing that the locker selected will be even does not impact the probability, so the
events are independent.
Out of the 5 even-numbered lockers, only three have gift cards, so P(A|B) =
Alternate Solution:
3
P(A) =
5
1
P(B) =
2
3
P(A and B) =
10
3
P(A) × P(B) =
10
Since P(A) × P(B) = P(A and B), then the two events are independent.
S.CP.A.3 Item 2:
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
Are the events “freshman” and “attended the spirit dance” independent events? Explain your
reasoning.
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Sample Solution:
300 1
P(freshman) =
=
1200 4
P(freshman|attended the spirit dance) =
240 1
=
960 4
So, the two events are independent.
Alternate Solution:
300 1
P(freshman) =
=
1200 4
P(attended spirit dance) =
960 4
=
1200 5
1
5
240 1
P(freshman and attended spirit dance) =
=
1200 5
Since P(A) × P(B) = P(A and B), the two events are independent.
P(freshman) × P(attended spirit dance) =
S.CP.A.3 Item 3: (More Questions under S.CP.B.6 Item 1)
The Boosters organization had a variety of door prizes for the Senior basketball game. They
gave sweatshirts to the first 20 fans and they gave spirit towels to anyone wearing the school
colors. The table below represents the results of the door prize distribution.
Student
Adult
Total
Received sweatshirt
15
5
20
Did not receive
sweatshirt
305
105
425
Received spirit towel
255
35
290
Did not receive spirit
towel
70
85
155
Total
325
120
445
Are the events “won a spirit towel” and “student” independent events? Support your answer
using appropriate probability calculations.
Solution: If independent, P(won spirit towel|student) = P(won spirit towel), P(won spirit
towel|student) = 255/325, P(won spirit towel) = 290/445. Since these probabilities are not equal,
the events are not independent.
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
S.CP.A.3 Item 4: (More Questions under S.CP.B.6 Item 3)
On October 3, 2013, two lion cubs were born at the Maryland Zoo in Baltimore. Over the next
few months, the zoo had a poll for the public to name the cubs. The results of the votes are
shown in the table below:
Male Voters
Female Voters
Total
Luke & Leia
12,122
8,382
20,504
Bart & Maggie
780
154
934
Kulu & Madoa
5,525
13,435
18,960
Lear & Circe
923
1,997
2,920
Total
19,350
23,968
43,318
Are the events “Luke & Leia” and “male voter” independent? Support your answer using
appropriate probability calculations.
Solution: If independent, P(Luke & Leia|male) = P(Luke & Leia), P(Luke & Leia|male) =
12122/19350, P(male) = 19350/43318. Since these probabilities are not equal, the events are not
independent.
Understand independence and conditional probability and use them to interpret data.
S.CP.A.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.A.5 Task
Write an example of two events that should result in P(A) = P(A|B). Explain what P(A) =
P(A|B) means in the context of your example.
Answers will vary. Responses should include: If the events A and B are independent, then the
probability that one of the events happens is not influenced by whether or not the other event has
happened (or is happening). An example might be the probability that a student plays an
instrument and the probability a student is above average height.
S.CP.A.5 Item 1
Jackie sometimes sleeps for more than eight hours and sometimes does not. She determines that
the events “I sleep for more than eight hours” and “I pass my math test” are independent.
Explain what this means in terms of the relationship between Jackie getting more than eight
hours of sleep and the probability that she will pass a math test in language that someone who
hasn’t studied probability will understand.
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Possible Solution: If “I sleep for more than eight hours” and “I pass my math test” are
independent, then the probability that one of the events happens is not influenced by whether or
not the other event has happened (or is happening). In this case, what Jackie is saying is that the
probability that she passes a math test is the same whether or not she sleeps for more than eight
hours.
S.CP.A.5 Item 2
Examine the following events:
Event A: “I get the flu.”
Event B: “I do not wear a winter coat”
If you determine that the P(A) = P(A|B), explain what this conclusion means in the context of the
problem.
Sample solution: If P(A) = P(A|B), then the events are independent. This means that the
probability that one of the events happens is not influenced by whether or not the other event has
happened (or is happening). In this case, the probability that you will get the flu is the same
regardless of whether or not you wear a winter coat.
S.CP.A.5 Item 3
Examine the following events:
Event A: “A student makes the soccer team.”
Event B: “A student attends a summer soccer camp.”
The soccer coach determines that the P(A) ¹ P(A|B). Explain what this conclusion means in the
context of the problem.
Sample solution: Since P(A) ¹ P(A|B), the events are not independent. This means that the
probability that one of the events happens is influenced by whether or not the other event has
happened (or is happening). In this case, the probability that a student makes the soccer team is
different based on whether or not the student attends a summer soccer camp.
S.CP.A.5 Item 4
Examine the following events:
Event A: “A student makes the honor roll.”
Event B: “A student eats breakfast every day.”
Would you expect P(A) = P(A|B)? Why or why not? Explain what your conclusion means in
the context of the problem.
Answers will vary. Students may argue that the events are independent (P(A) = P(A|B)) or are
not independent (P(A) ¹ P(A|B)). Focus on the students reasoning and whether they can
describe what their conclusion means in the context of the problem. For example, if they argue
dependence, students will state that breakfast has an influence on grades.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.