In the second book of the Lone Wolf series of game books, there happens to be a chapter describing a gambling game.
The game rules goes like:
- the player bets an (integer) amount of money
- the player chooses a number between 0 and 9
- a random number is rolled, between 0 and 9 (equiprobably), with the following effects:
- if the player guessed correctly, he wins 8 times his wager
- if the number the player picked was just before or after the roll (0 and 9 are adjacent for this purpose), he wins 5 times his wager
This is obviously a good game to play, as the expected gain is 1.1 times the wager. However, if the player goes broke, he loses.
Now, given a starting capital of $n$ coins, and a target of $t$ coins:
- is, as I suspect, the best strategy to only ever bet a single coin to reduce the probability of losing?
- what is the probability of reaching the target when playing with the optimal strategy?
- EDIT : and what's the distribution of the gains once the target is reached?
EDIT: as the best course of action seems to bet a coin at a time, it is possible to write a transition matrix. Then I suppose one could compute its limit when repeatedly squared, and have the answer. However, is there a nicer closed-form equation?
EDIT2: I wrote a transition matrix approximation (documented here), but I am still looking for a nice symbolic solution, if at all possible.