Fun Probability Problem

@Ian Fowler Exactly. I want the section that connects my two random picked points to include one or both of x=+-r.

I got your explanation of why it approaches infinity. Sounds reasonable, thanks!

Circle problem

I shall use k for "pi"

Area of large circle = kR^2

Area of small circle = kr^2

Area of ring = kR^2 - kr^2 = k(R^2 - r^2)

Choose 2 points P1 and P2 at random inside the large circle. What is the probability that the line joining P1 and P2 will cross the smaller circle. I will show 2 methods.

1) P1 is in the small circle. P = kr^2 / kR^2 = r^2 / R^2

P2 must lie in the ring. P = k(R^2-r^2) / kR^2 = (R^2 - r^2)/ R^2

Mulitply since both must happen

P(crossing) = r^2(R^2 - r^2) / R^4

We repeat the process for P2 in the small circle and P1 in the ring and get the same answer so we double.

P(crossing)

= [r^2/R^2] [(R^2-r^2)/R^2] *2

= 2r^2(R^2 - r^2) / R^4

2) P(crossing)

= 1 - both lie in the ring - both lie in the small circle

= 1 - [k(R^2-r^2)/kR^2][k(R^2-r^2)/kR^2] - [kr^2/kR^2][kr^2/kR^2]

cancelling out the "k"s

= 1 - [(R^2 - r^2)^2/R^4] - [r^4/R^4]

R^4 is the common denominator

= [ R^4 - (R^4 - 2r^2R^2 + r^4) - r^4 ] / R^4

= [R^4 - R^4 + 2r^2R^2 - r^4 - r^4] / R^4

= [ 2r^2R^2 - 2r^4] / R^4

factoring the numerator

= 2r^2(R^2 - r^2) / R^4

BTW you can still apply your analysis when r is fixed and R ---> inf

Now the chances are that both P1 and P2 will be in the ring and P(crossing) ---> 0

@Ian Fowler I understand your analysis, but it's not correct. If P1 and P2 both lie in the ring, they still can cross the smaller circle (they can also touch it):

Let φ=r/R. Then the line joining P1(x1,y1) and P2(x2,y2)

1) Definately never crosses if they both lie in the small circle

Probability of that happening = φ^4

2) Definately crosses if they lie in different areas (you considered it would be enough)

P = 2φ^2(1-φ^2)

3) We don't know/depend on the angle/distance from the centre/a lot of parameters

if they both lie in the ring

You may look at what I did in Desmos:

https://www.desmos.com/calculator/agcwvhxo2f

I tried to simulate "the line crossing the circle" as the problem of the system of equations having roots