So the other day I was watching a speedrun of the Legend of Zelda: Ocarina of Time (a speedrun is a playthrough of a game with the intent of beating the game as fast as possible). In one particular part of the game, the player is forced to follow around a gravedigger as he digs up various holes. There is one particular outcome that is desired (the "jackpot" of the game, essentially): a permanent health upgrade. There are also three undesirable outcomes that only give out money: a green rupee (the least valuable prize, pretty much $1), a blue rupee (a fairly desirable prize, say about $5), and a red rupee (a very desirable prize, say $20).
Here are the rules:
The chances of digging up a green rupee is 40%, a blue rupee 30%, a red rupee 20%, and the health upgrade 10%.
If the gravedigger has dug up eight green rupees already, and he would dig up a ninth green rupee this time around, he will instead dig up the health upgrade.
If the gravedigger has dug up four blue rupees already and he would dig up a fifth this time around, he will instead dig up a green rupee.
If the gravedigger has dug up two red rupees already and he would dig up a third this time around, he will instead dig up a blue rupee.
As a result, the maximum number of attempts to dig up the health upgrade is fifteen.
Example: four blue rupees and two red rupees have been dug up. If the gravedigger hit the 20% chance to dig up a red rupee on his seventh total attempt, he would instead dig up a blue rupee. However, four blue rupees have already been dug up, so he would actually dig up a green rupee.
After digging up the health upgrade, the game is over.
One: what is the probability that it will take a player the maximum number of tries to dig up the most desirable outcome (the health upgrade)?
Two: What is the expected average number of tries for the health upgrade?
Three: There exists two methods to play this minigame. First is the method described above. Second is that after the first dig, the player exits and reenters the area. This resets the green/blue/red rupee counter, but also allows the player to try another dig far faster. This introduces the potential for an infinite number of tries (the world record currently sits at about 100, which is pretty unlucky to say the least). Say it takes a player ten seconds between digs using the first method, and five seconds between digs using the second method. On average, which one will get you the health upgrade the fastest?
Disclaimer: I've got an answer to the first question and maybe the second, but I've got no clue for the third.