Average number of random stat increases to go from base to cap, where chance of increase reduces as stat nears cap

Question

Ok, so I have a game where you have 3 stats: agility, strength, and stamina. I'll avoid going into tons of detail about the game, so here's the gist of the math:

Each stat starts at 1. Every time you send a message, there is a random chance for a stat to increase by 1. The chance is calculated as follows:

First: The stat to try and increase is randomly selected. Equal probabilities.

Second: If the stat is at its cap (200), then nothing happens. No stats gained. Otherwise, the stat has a random chance equal to (200 - current stat) / 200 of increasing by 1.

What I want to know is, how many messages will it take, on average, for the stat to hit max? If you can tell me the same info with a starting stat of 200 each, and cap of 455 each, that would be lovely as well. If your formula applies to both, I can... probably use it? It's almost 2am, I'm kinda fried. Not sure what to tag. Anywho, thanks :) hope this is fun.

Answer 1

Okay, I think I figured this out on my own (after sleeping haha). I was really tired so I had to stupid-simple explain it to myself, and I'll write that here for fun and for the tired readers.

When you flip a coin, you have a 1/2 chance of getting heads. That means, for every 2 flips, you get 1 heads. 2 flips / 1 heads = 2 flips per head.

When you get a stat increase, you have a (200 - current stat) / 200 chance of getting the stat to increase by 1. So, for every 200 opportunities, you will get (200 - current stat) increases. 200 messages / (200 - current stat) stats = X messages per stat.

So, when my stat is 1, it's (200/199) messages to get it to increase to 2. When it's 2, it's (200/198). All the way up to when my stat reaches 199, and it will take (200/199) messages to get it to hit 200.

So, for one stat, the average number of messages to max it are:

(200/199) + (200/198) + ... + (200/2) + (200/1)

Which can also be written as:

The sum from i=1 to 199 of (200/i)

And this is easily changed for a start of 200 and a goal of 455 (I wrote 405, I meant 455, my bad):

The sum from i=1 to 454 of (455/i)

In the end I just plugged these into wolframalpha, I'm sure there's a formula for the sums but I'm too tired to remember that or my answers. Anyways, thanks for reading :) lmk if you have questions.