I came up with this problem on my own, and I can't figure out how to solve it.
You are a hero fighting an evil overlord. At the beginning of these events, you have 0% chance of defeating the overlord in a fight. You take one day to train, and at the end, you have a 10% chance of defeating the overlord in a fight. However, the night of the first day, there is a one in ten chance that the overlord will take over the world, and then it will be too late to stop him. On the second day, you can either go to fight the overlord, or train some more. At the end of the second day, you now have a 20% chance of defeating the overlord in a fight. However, if you fight the overlord, and lose, then there will be no one left to oppose him, and the world is lost. The night of the second day, there is another one in ten chance of the overlord taking over the world. Every day that you train, your chances of defeating the overlord become 10% better. However, you also run the risk of him taking over the world every night. What is the best number of days to train before fighting the overlord?
I'm trying to solve this without using a decision tree. The only thing I came up with was using an equation, like this: Y = (9/10)x * (1/10) * x
But I don't think that it was right, because it said the 9th and 10th days were the best, as they were identical to the decimal extent of my graphing calculator, at approximately a 35% chance for each. I don't think the solution should go beyond the 10th day, as there is a 100% theoretical probability that asking for a single 1/10 chance to succeed in ten tries.
Thank you now for your time, and in advance for your words.