I had a problem: I wanted one player's victory thresholds to be hidden from the other players, but didn't want that player to be hobbled by pure dumb luck.
The game was Westphalia, and the player was Austria. The way Austria works is that they start with a bunch of territory and a bunch of authoritarian prerogatives, and over the course of the game they're ceding territory and giving up some of those prerogatives in order to satisfy the demands of their enemies and safeguard the victories of their allies. The trick is that they can't give away too much, or they lose the game, and the other players can't know exactly how much Austria can or can't give away, otherwise they'd be able to math it out.
Hence, the thresholds on their goal cards: give away too much territory, you lose the game, give ground on too many of your prerogatives, you lose the game, and of course, hold too much debt, you lose the game. These cards also come with a Victory Point total, awarded to Austria (and the other HRE powers) should the game go to scoring: the harder the goals on Austria's card are to meet, the higher the VPs.
Originally Austria just shuffled their little deck of goal cards, pulled the top one off the deck, and that was that: here's your goal, hope you meet it. With six cards, that meant there would be an equal chance (roughly 16%) of the player getting the easiest goal (1 VP, doable, with cooperation) and the hardest goal (6 VP, mathematically possible, but unlikely in practice). In the latter case, especially with a new player, the game would be over before it began.
I briefly - very briefly, it never made it to the table - considered just letting Austria look at all six cards and then pick their goal, but since scoring happens only rarely with competent play, the higher VPs would hardly entice Austria to do more work than they needed to. They would always pick the easiest goal, in which case it was no longer hidden from the other players.
The next logical step of course would be for the player to draw two cards, and pick one. Now, the one they would pick would of course be the easiest goal, with the lowest VP value. But because they'd be drawing two cards, the probabilities shifted. There are fifteen possible two card "hands" that can come up with a deck of six cards, and the easiest 1 VP goal card comes up in five of them - 33% of the time. The 2 VP card would be the lowest card out of a pair in four hands - roughly 27% of the time. For 3 VP, the probability is 20%, and for 4 VP, 13%. The worst possible outcome - the player stuck with a choice between a 5 VP or a 6VP, where they would likely choose the fiver - could only occur in one hand out of fifteen, about 7% of the time.
This felt a little better, but I decided to push it further, and in the finished game, the player draws three cards and chooses one. Again, as a matter of course they're going to choose the easiest card. There are twenty possible hands in this case, and ten of them - 50% - would contain the 1VP card. Six hands - 30% - would see the 2VP card as the easiest goal possible, while three hands - 15% - would make the 3VP card the easiest. That leaves only one hand remaining - containing the 4, 5, and 6 - which would happen only 5% of the time.
That's a 95% chance your goal will be one of the first three cards. This gives the other players a general idea of the expected, normal range, but keeps it just a little fuzzy. Austria is more likely to get a goal that most players can pull off, which lets them ease up on the calculations, and concentrate on their other victory condition - securing the victories of Spain and Bavaria.
Math is not my strong suit; I struggle often with doing simple arithmetic in my head, as folks who play train games with me can attest. And I used to have a lot of anxiety about that, especially once I started designing board games. Many of the designers I admired had backgrounds in mathematics, and there was this general impression that game design - particularly of the euro variety - consisted of delicately-balanced resource conversion rates on spreadsheets. Wargames, particularly of the traditional hex-and-counter odds-ratio variety, also have a heavy mathematical bent. It seemed everywhere I turned, I was reminded that I was inadequate, if not innumerate.
I got over that eventually - do enough games and you start to convince yourself you might just know what you're doing. Part of that however was learning to use math as a tool rather than a discipline. I don't need to have a complete understanding of fluid dynamics to know how to use a hose. To figure out the probabilities and percentages of "draw three goal cards from a deck of six and choose the easiest one", I didn't fall back on my knowledge of combinatorials, which is practically nonexistent: I literally just counted them out on my fingers and my toes - good thing there were only twenty hands to count! - and wrote them down.
And here's the thing: it works either way. It might be a little slower, but it comes out the same in the end. Heck, if I hadn't told you about it, you would probably think I knew how to solve quadratic equations or something. My point here is, you don't need to be a math whiz to apply it to the craft and art of game design.
One final note about this particular practical problem. Since the game's release, some have asked me why I let Austria have a choice at all, since they're just going to choose the easiest hurdle every time. And that's a fair question. While there are some very specific situations where a high-level player might choose the hardest goal available, mostly it was just that I figured it wouldn’t hurt: if someone wants to make it harder on themselves, if that makes it more interesting for them, they can do that.