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Help with Probability and Integration


I recently discovered the website and it has helped me regain some of my math abilities that I lost. I am struggling with a small problem.

Let's consider a 2-D coordinate system Oxy. I have a light source at O which sends out at ray at random directions. I have a line at x=1 that goes from y=-1 to y=1. I want to calculate the probability that a ray hitting the line.

Here is how I thought of doing it. \(\theta\) is uniformly distributed on \([-\pi, +\pi]\) so probability density is \(f(\theta) = \frac{1}{2\pi}\) , right?

So the probability that I am looking for is : \(P =\int_{-1}^{1} f(arctan(y)) dy \) . We have \( y = tan(\theta) \) which gives us \(dy = (1 + tan^{2} (\theta)) d\theta \) When we substitute we get: \( P = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} f(\theta) (1 + tan^{2} (\theta) ) d\theta \) which gives us \( P = \frac{1}{2\pi} \) which I know is wrong since we are suppose to get 0.25.

So where did I go wrong?

Note by Alperen Aydin
3 weeks, 4 days ago

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