Imagine, if you will, a knight. He is a brave knight and is on a quest, exploring a haunted castle. He must make his way through five doors to reach the throne room of the castle.

Our knight, while brave, has been placed under a curse by an evil wizard. He has become easily distracted. He has only a one half chance of making it through the first door of the castle, and a one half chance of getting distracted by something in the foyer and spending a minute fiddling around. After a minute has passed, he can try again to continue on his way, but his distraction has weakened his focus, and he now only has a 1 in 3 chance of proceeding. And if he fails this time? 1 in 4. Each time he becomes distracted, his chance of continuing his mission decreases. When he does pass through that first door, his chance of continuing stays the same. So, for example, if he passed through the first door with a 1 in 5 chance, his chances of getting through the second door start at 1 in 5.

The question is, how many chances does our knight need to get through five doors? There is a 1 in 32 chance that he will pass through all of five gates by getting his half chance of proceeding each time. But once he rolls "falsely", his chances of proceeding can quickly deteriorate. The problem with trying to compute an answer to this problem is that the chances of not proceeding, of being distracted, can stretch out to infinity. There may not be a finite answer to the question. Someone can write a program to model our distracted knight, and run it through hundreds or thousands of iterations, to try to find an "average" answer, but those average answers don't deal with the outlier cases, where the amount of tries get larger and larger until the chance of proceeding becomes too miniscule to matter. Trying to find a systemic answer by logical deduction seems to be even more difficult to do.

Notice that both the numerator and denominator of the problem, as well as the number of rooms, are arbitrary for the sake of simplicity. We could begin with a 31/78 chance, in a castle of 14 rooms, and increase the denominator to 79 upon failure. However, the more simple case I presented starting with a 1 in 2 chance should be solved first, before trying to figure out a general formula.

While the lore behind this mathematical puzzle might be frivolous, the principle of compounding disadvantages is important in different fields of science. Our distracted knight might also be a seed that, if it germinates first, will easily proceed to grow larger, but if it germinates later, will quickly be shaded out by other trees. Or it could be a worker whose first job had low pay and a low path to advancement, and who then is at a disadvantage economically.

Feel free to send me any ideas about how to predict our distracted knight's journey in a comment, or feel free to append another writeup to this node.