Problem 12: Energy Crisis
In the collectible card game "Magic", players get "land" cards which they can play to generate "mana," which represents magical energy that can be used to play other cards which represent magical spells. Each land card can be "tapped" for one mana per turn, and different cards cost different amounts of mana to play. If you get lots of cards but not enough mana to play any of them that is called being "mana screwed." Unhinged, a "joke" Magic card set, contains a card called "Mana Screw", which allows the user to (as many times as he wants to) spend one mana to flip a coin, and if he wins the coin flip he gets two mana (for a net gain of one.)
Suppose that a player has X mana, and he has a spell that he wants to play that costs Y mana (where Y is greater than X). He uses Mana Screw's power repeatedly until he either (a) runs out of mana, or (b) gets enough mana to cast his spell. What is the probability that he will get enough mana to cast the spell.
There exists another Magic card called Krark's Thumb (this is not a joke card) that allows the user to, whenever he is called upon to flip a coin, flip two coins and ignore one. Thus if a player used Krark's Thumb he would have a 3/4 chance of winning each coin flip, rather than 1/2. Suppose that such a player had one mana, and wanted to play Gleemax, another "joke" card that costs 1,000,000 mana. He again continually used Mana Screw's power until he either ran out of mana or got enough to play Gleemax. What is the probability of success? (You can take the limit as Gleemax's cost approaches infinity.)
(Hint for both problems: Let f(x) be the probability of success starting from x mana, and find a recurrence relation for x.)
The solution is here.