Help with Probability and Integration

Hello,

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?

Easy Math Editor

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$