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Algebra I: Strand 2. Linear Functions; Topic 10. Probability; Task 2.10.4
TASK 2.10.4: WHAT’S THE CHANCE? — AREA
Solutions
The Happy Days High School math club plans on holding a “MIF” (Math Is Fun) night
with contests and prizes. They have created some dart games and now they need to
determine the chances for winning on each game to decide the best game for them to
offer.
The first proposed game looks like the following graph grid:
1. What is the probability of a student hitting the area represented by the inequality
y > x? Explain how you determined this.
1/2 the area of the grid is above the line so the probability is 1/2 or 50%
Another member of the games committee said too many people would win with that
grid and he proposed this graph grid.
2. Write an inequality to represent the area above the line. What is the probability of
a student hitting that area? Explain how you know.
y > 2x Half of the area of the grid is above the line; therefore, the probability a dart will
hit above the line is 50%.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.
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Algebra I: Strand 2. Linear Functions; Topic 10. Probability; Task 2.10.4
3. Here are two more graph grids. Write the inequality that represents the area
above the line for each graph and determine the probability of a dart hitting that
area.
y > 3x probability 50%
y > 6x probability 50%
4. Compare and explain the probability of hitting the area above the line in each of
the graph grids the students have been considering.
Any line that passes through the origin divides the graph grid into two congruent areas,
so the probability of a dart hitting above the line is 50%.
5. The developers decided that for each of the previous graph grids the probability a
student would hit the area above the line was too great. Another student put this
graph up for consideration. Write the inequality to describe the area above the
line and another inequality to describe the area below the line. Determine the
probability for each inequality.
y > 2x + 4; probability 40%
y < 2x + 4; probability 60%
NOTE: Students can compute the areas of the two trapezoids formed by the edges of the
grid and the separating line segment by using the formula. Compare each of the
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.
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Algebra I: Strand 2. Linear Functions; Topic 10. Probability; Task 2.10.4
trapezoid areas to the area of the total grid (20 x 20 = 400 sq units) to find the geometric
probability.
6. Does the position of the y-intercept in the graph above make a difference in the
area above the line? Explain what is happening to the area as the y-intercept is
shifted up or down the axis.
A line that passes through the origin divides the graph area into two congruent areas. As
the y-intercept increases the line divides the area of the grid into two unequal areas. As
the y-intercept increases the area above the line becomes smaller and the area below the
line becomes larger.
7. The game developers finally decided on a two part game. To win a big prize, first
the student would have to hit the area y > x, and then on a second try the student
would have to hit the area y > 2x + 4. What is the probability a student will win a
big prize? Justify your answer.
The probability of hitting the area above the line y > x is 1/2. The probability of hitting
above the area y > 2x + 4 is 2/5. The probability of both is 2/5 of 1/2 or 1/5, 20%.
8. Some textbooks explain solving compound probability with the rule “multiply the
probability of one event by the probability of the second event”. Illustrate why
this works.
Sample solution #1 For every ten coke bottles only four are put in a recycle bin, of
those in the recycle bin only 1/2 are actually recycled.
4 out of 10
1/2 of 4 out of 10. The probability is a
part to whole relationship. Dependent
probability is taking part of a part.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.
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Algebra I: Strand 2. Linear Functions; Topic 10. Probability; Task 2.10.4
Sample solution #2
One out of every 4 books checked out of the library is fiction. Of the fiction books
checked out, 1/2 are science fiction. What is the probability that the next book checked
out of the library will be science fiction?
1 out of 4
Half of the 1 out of 4
Half of 1 out of four is 1 out
of 8
Teaching notes
Students may ask about the probability of landing on the line itself. This may be an
opportune time to reinforce the geometry concept that a line has no width or height.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.
4
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Algebra I: Strand 2. Linear Functions; Topic 10. Probability; Task 2.10.4
TASK 2.10.4: WHAT’S THE CHANCE? — AREA
The Happy Days High School math club plans on holding a “MIF” (Math Is Fun) night
with contests and prizes. They have created some dart games and now they need to
determine the chances for winning on each game to decide the best game for them to
offer.
The first proposed game looks like the following graph grid:
1. What is the probability of a student hitting the area represented by the inequality
y > x? Explain how you determined this.
Another member of the games committee said too many people would win with that
grid and he proposed this graph grid.
2. Write an inequality to represent the area above the line. What is the probability of
a student hitting that area? Explain how you know.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.
5
6
Algebra I: Strand 2. Linear Functions; Topic 10. Probability; Task 2.10.4
3. Here are two more graph grids. Write the inequality that represents the area
above the line for each graph and determine the probability of a dart hitting that
area.
4. Compare and explain the probability of hitting the area above the line in each of
the graph grids the students have been considering.
5. The developers decided that for each of the previous graph grids the probability a
student would hit the area above the line was too great. Another student put this
graph up for consideration. Write the inequality to describe the area above the
line and another inequality to describe the area below the line. Determine the
probability for each inequality.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.
6
7
Algebra I: Strand 2. Linear Functions; Topic 10. Probability; Task 2.10.4
6. Does the position of the y-intercept in the graph above make a difference in the
area above the line? Explain what is happening to the area as the y-intercept is
shifted up or down the axis.
7. The game developers finally decided on a two part game. To win a big prize, first
the student would have to hit the area y > x, and then on a second try the student
would have to hit the area y > 2x + 4. What is the probability that a student will
win a big prize? Justify your answer.
8. Some textbooks explain solving compound probability with the rule “multiply the
probability of one event by the probability of the second event”. Illustrate why
this works.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.
7