2. On this problem, give exact answers Xing fractions. 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 in this problem is to compute P(X < (3/4) given Y < (1/2)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 = (1/2)X in the sample space. Label the points where this line intersects the corners or sides of the unit square. (c) Compute P(Y < (1/2)X) by finding the area of the triangular region underneath the line from part (b) and dividing by the area of the sample space. (d) Draw the graph of the vertical line X = (3/4) in the sample space. Label the points where this line intersects the sides of the unit square, and the point where this line intersects the line Y = (1/2)X from part (b). (e) Sketch the region where X < (3/4) and Y < (1/2)X. Use shading or coloring. (f) Compute P(X < (3/4) and Y < (1/2)X) by finding the area of the triangular region from part (e) and dividing by the area of the sample space. (g) Use your answers to parts (f) and (c) to compute the conditional probability P(X < (3/4) given Y < : (1/2)X).
2. On this problem, give exact answers Xing fractions. 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 in this problem is to compute P(X < (3/4) given Y < (1/2)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 = (1/2)X in the sample space. Label the points where this line intersects the corners or sides of the unit square. (c) Compute P(Y < (1/2)X) by finding the area of the triangular region underneath the line from part (b) and dividing by the area of the sample space. (d) Draw the graph of the vertical line X = (3/4) in the sample space. Label the points where this line intersects the sides of the unit square, and the point where this line intersects the line Y = (1/2)X from part (b). (e) Sketch the region where X < (3/4) and Y < (1/2)X. Use shading or coloring. (f) Compute P(X < (3/4) and Y < (1/2)X) by finding the area of the triangular region from part (e) and dividing by the area of the sample space. (g) Use your answers to parts (f) and (c) to compute the conditional probability P(X < (3/4) given Y < : (1/2)X).
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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![2. On this problem, give exact answers Xing fractions.
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 in this problem is to compute P(X < (3/4) given Y < (1/2)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 = (1/2)X in the sample space. Label the points
where this line intersects the corners or sides of the unit square.
(c) Compute P(Y < (1/2)X) by finding the area of the triangular region underneath
the line from part (b) and dividing by the area of the sample space.
(d) Draw the graph of the vertical line X = (3/4) in the sample space. Label the
points where this line intersects the sides of the unit square, and the point where
this line intersects the line Y = (1/2)X from part (b).
(e) Sketch the region where X < (3/4) and Y < (1/2)X. Use shading or coloring.
(f) Compute P(X < (3/4) and Y < (1/2)X) by finding the area of the triangular
region from part (e) and dividing by the area of the sample space.
(g) Use your answers to parts (f) and (c) to compute the conditional probability
P(X < (3/4) given Y < : (1/2)X).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b2b0246-74c5-427f-bbae-6a253749f017%2F1f6b9caa-736e-48ac-9b69-6be26a5a794a%2F5mfh1kc_processed.png&w=3840&q=75)
Transcribed Image Text:2. On this problem, give exact answers Xing fractions.
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 in this problem is to compute P(X < (3/4) given Y < (1/2)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 = (1/2)X in the sample space. Label the points
where this line intersects the corners or sides of the unit square.
(c) Compute P(Y < (1/2)X) by finding the area of the triangular region underneath
the line from part (b) and dividing by the area of the sample space.
(d) Draw the graph of the vertical line X = (3/4) in the sample space. Label the
points where this line intersects the sides of the unit square, and the point where
this line intersects the line Y = (1/2)X from part (b).
(e) Sketch the region where X < (3/4) and Y < (1/2)X. Use shading or coloring.
(f) Compute P(X < (3/4) and Y < (1/2)X) by finding the area of the triangular
region from part (e) and dividing by the area of the sample space.
(g) Use your answers to parts (f) and (c) to compute the conditional probability
P(X < (3/4) given Y < : (1/2)X).
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