# What is the probability that the angle is a right angle?

Let's say you have a square, say $$ABCD$$, with side length 1. A point $$P$$ is randomly chosen from inside the square, with any point being equally probable of being chosen. What is the probability that the angle $$\angle APB$$ is a right angle?

This looks simple, and I do this problem for the case of $$\angle APB$$ being acute or obtuse, and that the set of all points satisfying the condition above is a semicircle inside the square centered at the midpoint of $$AB$$. How do you do this, or is it impossible?

• My guess: probability of being a right angle (or any partticular angle) is $0$. – herb steinberg May 5 '19 at 21:59

The probability is zero. You've correctly identified the set of all points satisfying the desired condition -- the locus is a semicircle. In the acute or obtuse case, you were able to find the locus, and it turned out to have positive two-dimensional area. You were able to use this to calculate a probability. For the right angle case, the locus isn't "two dimensional" -- it's one dimensional. Thus its two-dimensional area is $$0$$, so the probability is $$0$$. Intuitively, it is impossible to randomly pick a point exactly on that locus.
As for the angle being obtuse, the associated area is a semi-circle centred at the midpoint of the line $$AB$$ with radius $$0.5$$ hence its total area is $$\frac12\pi(0.5)^2=\frac18\pi$$ as the total area of the shape is $$1\times1=1$$ we have $$\text{probability }\angle APB\text{ obtuse}=\frac18\pi$$ $$\text{probability }\angle APB\text{ acute}=1-\frac18\pi$$
• Yes. It is also 'impossible' to choose any single real number randomly from the interval $[0,1]$. You should look into measure theory. – Peter Foreman May 5 '19 at 22:04
• @Jithinash - in probability (especially with continuous distributions), a probability of $0$ does not mean impossible or never, and a probability of $1$ does not mean sure but instead almost sure – Henry May 5 '19 at 22:06