Steven2010-01-04 05:48:35

I’m stuck with just part of a problem (just the parts that I’m stuck on are

listed below) and was looking for some help

Poor Bobby wants to gamble, but he has no money. Luckily, his benevolent

father offers to give him five dollars to play quarter-dollar slot machines,

saying “if you ever go bust, you can return to me for another five-dollars

worth of quarters, but if your total holdings ever exceeds ten dollars, you

may keep five dollars worth of quarters but return everything else to me.”

Suppose Bobby’s slot machines only take quarter-dollar bets (i.e. one

quarter per spin) and only two outcomes are possible for each spin: (i) a

return of nine quarters with probability .1 and (ii) no return with

probability .9.

i. Compute the steady state probabilities for each recurrent

class above.

ii. What is the expected number of quarters in Bobby’s stash in steady

state?

I easily got the following equations but have no idea what to do next to get

the solutions to i and ii.

Let N = {1,2,3…,47} be the number of quarters that Bob has.

For 0 < n < 40
P(n , n+8) = .1
P(n , n-1) = .9
For n >= 40

P( n , 20 ) = 1

For n = 0

P ( n , 20 ) = 1.

Ken.pledger2010-01-04 05:48:47

Have you formulated the problem properly? When you say “Let N =

{1,2,3…,47} be the number of quarters that Bob has,” a good question is

“When?” Some of your transitions between successive states involve

gambling on machines, and some involve transactions with that highly

indulgent father. How about combining them?

I suggest you look at the state just before each foray into the

casino, labelling it by the number of quarters which Bob then has, viz 1,

2, 3, … or 40. Each transition involves a gamble plus possibly a

transaction with Dad. Then it seems to me that

for 2 =< n =< 40, P(n, n-1) = .9
P(1, 20) = .9
for 1 =< n =< 32, P(n, n+8) = .1
for 33 =< n =< 39, P(n, 20) = .1.
Have I got that right? If so, do you know enough of the theory to
move ahead? If you preferred, you could get a different formulation by
looking at the state just after each gamble instead of just before.
Ken Pledger.

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