Legends of Idleon MMO - Rarity Chances for the Arcade

Ever wondered how rare it is for a Arcade ball to end up in a specific category (after travelling through all these portals)?
Well, I don't know why I did this, but I've calculated it all out. Here are the results:

• gray ~63,680% [1 in 2]
  
• bronze ~29,581% [1 in 4]
  
• silver ~5,310% [1 in 19]
  
• gold ~1,110% [1 in 90]
  
• purple ~0,271% [1 in 370]
  
• red ~0,048% [1 in 2082]
  
• blue ~0,001% [1 in 93669]

One ball is worth around:

• 17,7 money - multiplicate that times your lvl
  
• 2,48617 Arcade shop money
  
• 0,21163 gems
  
• and has a 18,74% chance to get another ball (plus dice rolls!)

This is how rare a specific drop is:

• frag [1 in 16]
  
• golden food [1 in 54]
  
• dice roll [1 in 116]
  
• blue candy [1 in 170]
  
• item (silver category) [1 in 325]
  
• red candy [1 in 1300]
  
• balloon (gold category) [1 in 1830]
  
• stone [1 in 4617]
  
• green candy [1 in 6038]
  
• item (purple category) [1 in 6155]
  
• obols (purple category) [1 in 7386]
  
• yellow-gray candy [1 in 18465]
  
• 3 balloons (red category) [1 in 26020]
  
• obols (red category) [1 in 26020]
  
• pink candy [1 in 29736]
  
• stamp [1 in 60373]
  
• card [1 in 69385]
  
• hat [1 in 104077]

So out of 100 balls, results may vary greatly, but in general you can expect about:

• 1770*lvl money
  
• 249 Arcade shop money
  
• 6 frag
  
• 2 golden food
  
• 1 blue candy (59% chance)
  
• 1 dice roll (86% chance)
  
• 21 gems
  
• 19 balls

This assumes that each ball has a probability of 1/3 to hit one of the three gray pins at the top left corner and then has a 50-50 chance to move one step downwards either to the left or to the right. Also, when a ball is on the far left playfield, it is assumed to have a 100% chance to move to the right.
This basically lets you determine the probability of a ball landing in a specific category when it starts from the initial point or one of the portals.

But landing in a specific category does not mean that the ball ends up in it - Since a ball has a chance to go through a portal...
So I calculated in steps - first, I took all the balls which land in a category and miss the chance to go through a portal, then I looked at the remaining balls and figured out what would happen to them.

Difficulty arises once you realize that the arcade game can theoretically go on forever - since a ball landing (not ending up!) in the gray category has a chance to go to the bronze portal and has a chance to land in the gray category from there on once again. Luckily it is becoming less and less likely for a ball to go through another portal. During my calculations, I only took balls into acount which end up going through at most 15 portals - which is sufficient in 99,9999999929305% of all cases.

Don't rely on my results - although I checked some of these, I might have made calculation errors.