Earlier errors are now fixed. I hope there are no more.
Question: Two points on a circle of radius 1 are chosen at random. Find the probability that the distance between the two points is at most 1.
I have spent a long time on this, I think this answer is correct.
Orient your circle so that the interval joining the 2 points that you choose is above and parallel to the diameter.
The green semicircle is the top of the original circle.
If I draw 1 unit intervals paralel to the diameter how much of semicircle will not be included.
I needed a curve 1 unit to right of the original and the area in the bricked in area indicatesw the left over.
So the prob of the points being 1 unit or less apart is the clear green area over the whole green semicircle.
Area of Simicircle = π2
Area of the bricked sector sector =
60360∗π=π6
Area of bricked segment
=π6−12∗1∗√12−0.52=π6−12∗√32=π6−√34=2π−3√312
Total bricked area
=2π12+2π12−3√312=4π−3√312
Area on non-bricked section
6π12−4π−3√312=2π+3√312
Probability that the points will be less than or equal to 1 unit apart
=2π+3√312÷π2=3√3+2π6π≈61%
Hopefully I have now fixed all the errors.
