Luck Quantification

 It's surprisingly easy to quantify luck, if you have well-defined probabilities. It's linear.

 Say you have a 50% chance and a 75% chance. There's four permutations, and it goes up quadratically with more entries, but you don't need to care about that. You have a total of 125%. For your bad luck, a total of 75%. Each actual event counts for 100%, and you can just add those up too. Compare the total percentage to expected percentage, and you can calculate how lucky or unlucky you were in terms of profit. It's a way of getting around trying to calculate exact percentage likelihoods and stuff like that.

 It's important in real life for a couple reasons. If you get lucky early you may want to lock it in and bow out of the risks. You can also quickly work out if something is far from the norm (even estimated probabilities are good enough) and not strategize as if it's normal. 

 It's important in videogames. If a player is getting very unlucky, that's usually a bug. In any kind of social or competitive game, getting very lucky is also a bug. The game can quickly and easily track their luck using this sort of thing, and cap the extremes. 

 Imagine it's loot drops from a boss. You do ten pairs of this, so 1250%, you should have 12 or 13 of whatever you're trying to loot.  This captures all sorts of permutations, such as lots of the 50% and few of the 75%, or like one 50% and 12 of the 75%s. If you have something wildly different, then you know you got very lucky or unlucky. 

 

 It also reminds you that one coin toss is extremely luck-driven. You expect 50%, and you get either 0% or 100%. On one toss, you're always wildly lucky or wildly unlucky. With two tosses, there's still a very high chance of being wildly lucky or unlucky: 50%. 25% of 0%, and 25% of 100%. If you're betting a lot on only one roll, it's best to plan on a loss all the way up to 95% / 5% odds or even higher in some cases. 

 Pedantism: for 50 and 75, the odds of getting one of them are 0.5 * 0.25 + 0.5 * 0.75 = 50%, the odds of getting neither are 0.25 * 0.5 = 12.5%, and the remainder is the odds of getting both, 37.5%. 50 + 37.5*2 = 125. Doing luck quatification the above way skips this combinatoric explosion.

 P.S. There's a clever and easy way to cap player luck, by generating a deck of rolls instead of rolling on the fly. If e.g. the player is missing or critting way too often, the deck can shuffle the probability card back into the deck instead of playing it. This way the statistics aren't fudged overall - the player still misses just as often - but it avoids excessive streakiness which is a feature of true randomness.
 That said the game should be designed so the player can't just get killed by random streaks, rather than making it so painful it needs to adjust.