Let (X,Y) be a random point in the unit square, [0, 1] x [0, 1], with the uniform probability. That is, X is a random number between 0 and 1, and Y is a random number between 0 and 1. Your overall task is to compute P (Y < X and Y> (3/5)X). (a) Draw the Cartesian plane (make sure there is plenty of room in the first quadrant), and draw and label the axes. Sketch the sample space, [0, 1] x [0, 1], in the first quadrant. (b) Draw the graph of the line Y = X in the sample space. You may also want to label the points where this line intersects the unit square. (c) Draw the graph of the line Y = (3/5)X in the sample space. Hint: This line starts at the origin (the lower-left corner of the sample space) and intersects the unit square somewhere along the right side. You may also want to label the points where this line intersects the unit square. (d) Sketch the region where Y < X and Y> (3/5)X. Use shading or coloring. (e) Notice the sample space has been divided into three regions: the event region you sketched in part (d), and two triangles that form the complement of the event. Find the areas of each of these two triangles. (f) Use your answer to part (e) to find the area of the event region. (g) Use your answer to part (f) to find the probability of the event.
Let (X,Y) be a random point in the unit square, [0, 1] x [0, 1], with the uniform probability. That is, X is a random number between 0 and 1, and Y is a random number between 0 and 1. Your overall task is to compute P (Y < X and Y> (3/5)X). (a) Draw the Cartesian plane (make sure there is plenty of room in the first quadrant), and draw and label the axes. Sketch the sample space, [0, 1] x [0, 1], in the first quadrant. (b) Draw the graph of the line Y = X in the sample space. You may also want to label the points where this line intersects the unit square. (c) Draw the graph of the line Y = (3/5)X in the sample space. Hint: This line starts at the origin (the lower-left corner of the sample space) and intersects the unit square somewhere along the right side. You may also want to label the points where this line intersects the unit square. (d) Sketch the region where Y < X and Y> (3/5)X. Use shading or coloring. (e) Notice the sample space has been divided into three regions: the event region you sketched in part (d), and two triangles that form the complement of the event. Find the areas of each of these two triangles. (f) Use your answer to part (e) to find the area of the event region. (g) Use your answer to part (f) to find the probability of the event.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![Let (X,Y) be a random point in the unit square, [0, 1] × [0, 1], with the uniform probability.
That is, X is a random number between 0 and 1, and Y is a random number between 0 and 1.
Your overall task
is to compute P (Y < X and Y > (3/5)X).
(a) Draw the Cartesian plane (make sure there is plenty of room in the first quadrant), and draw and
label the axes. Sketch the sample space, [0, 1] × [0, 1], in the first quadrant.
(b) Draw the graph of the line Y = X in the sample space. You may also want to label the points
where this line intersects the unit square.
(c) Draw the graph of the line Y = (3/5)X in the sample space. Hint: This line starts at the origin
(the lower-left corner of the sample space) and intersects the unit square somewhere along the
right side. You may also want to label the points where this line intersects the unit square.
(d) Sketch the region where Y < X and Y> (3/5)X. Use shading or coloring.
(e) Notice the sample space has been divided into three regions: the event region you sketched in
part (d), and two triangles that form the complement of the event. Find the areas of each of these
two triangles.
(f) Use your answer to part (e) to find the area of the event region.
(g) Use your answer to part (f) to find the probability of the event.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd222e84f-10f7-4ebe-aaf6-917bab5ac2d8%2Fbf6a95b5-e6fc-492d-929b-91729d98dbc9%2Fcdsdlj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let (X,Y) be a random point in the unit square, [0, 1] × [0, 1], with the uniform probability.
That is, X is a random number between 0 and 1, and Y is a random number between 0 and 1.
Your overall task
is to compute P (Y < X and Y > (3/5)X).
(a) Draw the Cartesian plane (make sure there is plenty of room in the first quadrant), and draw and
label the axes. Sketch the sample space, [0, 1] × [0, 1], in the first quadrant.
(b) Draw the graph of the line Y = X in the sample space. You may also want to label the points
where this line intersects the unit square.
(c) Draw the graph of the line Y = (3/5)X in the sample space. Hint: This line starts at the origin
(the lower-left corner of the sample space) and intersects the unit square somewhere along the
right side. You may also want to label the points where this line intersects the unit square.
(d) Sketch the region where Y < X and Y> (3/5)X. Use shading or coloring.
(e) Notice the sample space has been divided into three regions: the event region you sketched in
part (d), and two triangles that form the complement of the event. Find the areas of each of these
two triangles.
(f) Use your answer to part (e) to find the area of the event region.
(g) Use your answer to part (f) to find the probability of the event.
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