Wim Benthem wrote: > > On 25 Nov 2000 01:54:27 GMT, email@example.com (Amy Shining Star) > wrote: > > >Suppose you start with $2. > > > >The game is as follows: > > > >You flip a fair coin. > >If you get a head, you gain $1. > >If you get a tail, you have a chance to pick a ball from an urn. > >With probability 0.6, you will get a green ball. Then you don't lose anything. > >With probability 0.3, you will get a blue ball. Then you will lose $1. > >With probability 0.1, you will get a red ball. Then you will lose $2. > > > >Repeat the game forever. If you have a negative balance, you must stop playing. > > > > > >What is the probability that the game will never stop? > >What is the probability that you will end up in debt (game stopped)? > > It's possible to find an exact solution. The probability that the game > will end is only dependent on the amount of money you have left. > lets define P(n) is the probability that the game will end if you have > n dollars left. > Now we can find a differential equation for P(n) > > P(n) = 0.5 P(n+1) + 0.3 P(n) + 0.15 P(n-1) + 0.05 P(n) > > as boundary conditions we have > P(-1) = 1 > P(-2) = 1 > and lim P(n) = 0 > n->oo > > if you can't solve the differential equation: look up linear > differential equations somewhere.
I've made some progress on this one, but I'm now stuck.