An archery target has rings of radius r, 2r, and 3r. Suppose an arrow hits the target randomly. What is the probability that an arrow hits the target will land in each ring?

area of target = π*(3r)^2 = 9πr^2
area of outer ring = 9πr^2 – 4πr^2 = 5πr^2
area of middle ring = 4πr^2 – πr^2 = 3πr^2
area of inner ring = πr^2

P[hit any particular ring] = area of that ring / area of target, so

P[hit inner ring] = 1/9
P[hit middle ring] = 3/9 = 1/3
P[hit outer ring] = 5/9

While using this particular setup, we need to make sure that the archery is positioned correctly to hit the target. If you are practicing while you are in the woods, you may need used outboard motors for the job.

I bought a couple medium sized ones, what is good to fill them with? How easy will arrow removal be?

If you fill them with sand, after a few well placed shots, the sand will spill out.
I do not feel that one could pack enough straw or hey in the bags to slow the arrow to a stop safely.
Stryofoam is not compact enough.
So I would forget the sandbags all together, unless you want to use them for a backstop or shield.
Buy a couple bails of hey or straw and put the target on them. The arrows can be removed with ease.

One

Meaning Olympic archery.

For Olympic style you have to use recurve. The recurves used for this are the take down variety with metal(aluminum) or carbon riser and carbon/wood or carbon/foam limbs.

I do suggest finding a range near you and get some pointers. Many ranges in the US provide beginner level equipment at no charge. This is a great way to get started properly. With Olympic style archery you need to start with very light bows or you will develop bad habits that are very hard to change later on. It can get very costly if you buy your own gear and have to climb up the poundage, most of the archers I train do the climbing with our equipment, then when they’re near their target weight (usually around 40lbs for men and 34lbs for women) they can buy their own stuff.

If you do decide to buy, here’s my advice. Get a nice riser, the riser can stay with you till the end of time. So get one that you like, with risers it’s all pretty much about the feel of the grip and how it behaves after the shot. Some risers like to jump out of your hand, others just roll. One is not better than the other, it’s just about what you prefer. Then of course there’s the look factor, is it available in the color that you like for example. You see, most risers made today will outshoot just about anyone alive.

I can’t afford a target temporarily, so I’m planning to use a tree in a forest near me… Is it not morally right to do so?

How about a dead tree, does the point still stand?

Dead tree stump target practice is a common thing to do. The soft wood stops an arrow without any damage to the arrow or tip. However a live tree is a hard wood and would break an arrow and cause damage to the tree. Just hike around till you find an old dead tree and use that.