> Because the monster battles are statistically independent, we can make a rough guess that about 1 in 9 players still won’t have a special item after 20 battles.
From a game design perspective, for some situations this can be a real problem that leads to players getting discouraged and quitting, which leads to loss of revenue. In the games I've made, I've often used a special trick for overcoming this problem. Rather than using plain random numbers, I conceptually use a shuffled deck of cards, where each player receives a different random seed for the shuffle and I maintain state on how many "cards" they've drawn so far. I put one "win" in the deck and some number of non-wins. This way it's still random, but if people keep at it they'll eventually get the thing that they want. With plain raw probabilities, you'll always have a certain number of customers that keep paying to try and keep failing. The random shuffle is a way better way to approach this from a customer service perspective.
This makes it easier to reason about from a game design and customer service perspective. I can say that no player will ever have to go through more than X attempts. I say it's a customer service issue because in many of my games the players feel like there's real money on the line, and so it can literally lead to customer service tickets, wailing, and gnashing of teeth.