Find pkby setting up and solving with a DIFFERENCE EQUATION (This is Discrete).
Two gamblers, A and B, make a series of bets, where each has probability 1/2 of
winning a bet, but A gets $2 for each win and loses $1 for each loss (a very favorable
game for A!).
Assume that the gamblers are allowed to borrow money, so they can
and do gamble forever.
Letpkbe the probability that A, starting with $k, will ever reach $0, for eachk=0.
Find
p
k.
This problem must be solved with a DIFFERENCE EQUATION (it is Discrete): Hint: We havepk->0 ask->8(you don't need to prove this, but it should make sense
since the game is so favorable to A, which will result in A's fortune going to8; a
formal proof, not required here, could be done using thelaw of large numbers
The solution should be written neatly in terms of the golden ratio.