Probability Chances that y>2x

[sciguy]

Homework Statement

Two numbers are chosen, (x and y) both in the interval [0,1]
What is the probability that y>2x

Homework Equations

N/A

The Attempt at a Solution

I'm new to all the probability stuff, so I'm still learning all the notation. So far, I know that if y is going to be greater than 2x, but still in the [0,1] range, then x must be in the range [0,.5]. I imagine the final probability would be (.5)*(?). I'm just not sure how you would find the (?). By intuition, I think it would be .5 as well, making the final answer .25, but I'm not sure why this is the case.

Thanks!
Jeremy

 

[KataKoniK]

Dear Jeremy,

To find the probability in this scenario, we can use the basic formula for probability: P(A) = number of favorable outcomes / total number of outcomes. In this case, the total number of outcomes is infinite, since x and y can take on any value in the interval [0,1]. However, we can still use integration to find the probability.

First, let's set up the equation for y>2x in terms of x and y:
y > 2x
y - 2x > 0

Now, we can graph this equation in the x-y plane to visualize the region where y is greater than 2x. It would look like a triangle with vertices at (0,0), (0.5,0.5), and (0,1). This triangle represents all the possible outcomes where y is greater than 2x.

The total area of the square [0,1] x [0,1] is 1, so the probability of y>2x would be the area of the triangle divided by 1. To find the area of the triangle, we can use integration:

∫∫(y - 2x)dydx, where the limits of integration are x = [0,0.5] and y = [0,1].

Solving this integral, we get the final probability to be 1/8 or 0.125. This means that the chance of y being greater than 2x in this scenario is 12.5%.

I hope this helps! Let me know if you have any further questions.