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Algebra I: Strand 2. Linear Functions; Topic 10. Probability; Task 2.10.3
TASK 2.10.3: WHAT’S THE CHANCE? — SLOPE
Solutions
1. If the numbers −2, 4, −6, and 8 randomly replace the letters in the coordinate
points (a, b) and (c, d), without repetition, what is the probability the slope is
positive? Justify your answer.
The possibilities are:
(−2 − 4)/(−6 − 8) (+) (−2 − 8)/(4 +6) (−)
(−2 + 6)/(4 − 8) (−)
(−2 −4)/(8 + 6) (−) (−2 − 8)/(−6 + 4) (+) (−2 +6)/(4 + 8) (+)
There will be 6 possibilities with each of the numbers in the “a” position which gives a
total of 24 in the sample space. All will produce a 3 out of 6 possibility of a positive
slope. The possibility of a positive slope is 50%.
The general statement would be:
b > d and a > c will result in positive slope
b < d and a < c will result in a positive slope
b > d and a < c will result in a negative slope
b < d and a > c will result in a negative slope
a. Would the probability that the slope is positive remain the same if all the
numbers were positive?
No
b. Would the probability that the slope is positive remain the same if all the
numbers were negative?
No
2. Is the probability of negative slope affected when 3 of 4 numbers in a sample
space are negative (without repetition)? Explain.
No; see #1.
3. Is the probability of a negative slope affected when 3 of 4 numbers in a sample
space are positive (without repetition)? Explain.
No; see #1.
4. How is the probability affected if there are 5 numbers in the sample space
(without repetition)? Explain.
There will be more possibilities in the sample space, but the result will be the same
answer as #1. Because only 4 numbers can be used at a time the only combinations
include the general result from #1.
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 2.10.3
Teacher notes
During the discussion of this task, make sure students see the “big idea”. There are only
4 possibilities. In computing the Δy only two numbers can be considered and if the first
is larger than the second, the result will be positive. The same holds for Δx. The four
possibilities then become:
L!S
S!L
L!S
S!L
=+
= (+)
= (!)
= (!)
L!S
S!L
S!L
L!S
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.
3
Algebra I: Strand 2. Linear Functions; Topic 10. Probability; Task 2.10.3
TASK 2.10.3: WHAT’S THE CHANCE? — SLOPE
1. If the numbers −2, 4, −6, and 8 randomly replace the letters in the coordinate
points (a, b) and (c, d), without repetition, what is the probability the slope is
positive? Justify your answer.
a. Would the probability that the slope is positive remain the same if all the
numbers were positive?
b. Would the probability that the slope is positive remain the same if all the
numbers were negative?
2. Is the probability of negative slope affected when 3 of 4 numbers in a sample
space are negative (without repetition)? Explain.
3. Is the probability of a negative slope affected when 3 of 4 numbers in a sample
space are positive (without repetition)? Explain.
4. How is the probability affected if there are 5 numbers in the sample space (no
repetition)? Explain.
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.