Mathematicians / CS types - Odds Algorithm

While spelunking across Wiki today I came across this https://en.wikipedia.org/wiki/Odds_algorithm explanation of the odds-strategy and the odds-algorithm. Which has to do with decision-making in a multiple trial activity.

Usually this stuff is right up my alley and this article seems clearly written. But this one leaves me utterly baffled. I’m hoping somebody can explain what I’m missing or confirm that I’ve got a point.

The problem statement is clear: How do you know when best to stop a repeated trial experiment given some *a priori *info about the likelihood of success of any given trial? The math of the I, P, Q, and R series are clear, bordering on obvious.

But the algorithm says in effect to start with the last trial and accumulate back towards the first to determine the optimal cutoff point. But the problem statement says you must process the trials only going forwards from the first with no backtracking. And for some of the examples, the P[sub]k[/sub] values inherently depend on those of the earlier entries.

The whole thing smells like a contradiction to me. Recast as a Monte Carlo simulation it’s obvious-ish how to compute it and how the answer will be valuable.

But as a more-or-less closed form series calc it seems to require knowledge you can’t have and to demand violations of its own constraints.

Where have I gone off the rails?

I’m no expert; I just read the Wikipedia article (and I didn’t even stay at a Holiday Inn express last night). But my understanding of the problem statement is that before any trials the probability of success for each trial, P[sub]k[/sub] is already known. Whether the trail will actually succeed is unknown, but the odds of success are known with certainty.

With the odds of success for each k known, then it’s possible to sum probabilities going backwards from n.