/sci/ - Science & Math

Reposting from /g/ since it seems more you guys area of expertise:

This is probably real basic but I got a math related kinda question I was wondering. There's this game I play where you can take your items to an NPC to be refined, which adds a + number to the item's name, and some added stats and other bonus stuff depending on the item. Anyway, going +1 to +4 is 100% chance, but going any higher has a 50% chance of going up, and a 50% chance of going down. If it goes down, there's also an additional 50% chance that the item will break, making it impossible to do further refinements. So basically it's 50% to succeed and go up, 25% to just go down, and 25% to go down and also break.

My question is, how would you calculate the overall probability to do something like going +4 to +8 before it breaks? Like you got the chance of it succeeding every time, but that's easy enough to calculate, but there's all kinds of other cases like going down and back up without breaking so it confuses me... Been a long time since I got out of school so I'm probably forgetting something simple though. Is there some fairly elegant way of calculating this? I attached a rough diagram of the possible outcomes for the first few repetitions to give a better idea of what I'm talking about.