Step by step solution to "Probability that x y ≤ 1 and x..."

High School
Geometry
High school geometry
Geometric Probability

Yair Valentine

Answered question

2022-08-10

Probability that

x y \leq 1

and

x / y \geq 2

Let x,y be any real numbers in the interval [0,2]. What is the probability that

x y \leq 1

and

x / y \geq 2

Answer & Explanation

ambivalentnoe1

Beginner2022-08-11Added 20 answers

Step 1
The probability space

[0, 2] \times [0, 2]

can be seen as a

2 \times 2

square with opposite corners at (0,0) and (2,2). Its area is 4. So given a region R in this square, the probability that a random point (x,y) is in R is simply the area of R divided by 4.
Step 2
So now you just have to figure out the area of the region where

x y \leq 1

and

x / y \geq 2

. Start by drawing a graph of the curves

y = 1 / x

and

y = x / 2

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