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Suppose there are vending machine that sells its goods for $3$. It's known that a third of the buyers use three coins of $1$, a third of the buyers use $2$ and $1$, and the last third use $5$. The machine start empty (without money). What it is the probability that it could give change after $n$ buyers?

Thank you!

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    What happens if someone who wants to use 5 tries to use the machine and it can't give change? (Or, more to the point, if this happens does it increment the number of buyers?)2011-01-19
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    @Yakov: You could find $1-P(\text{no change after} \ n \ \text{buyers})$.2011-01-19
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    If there are lots of 1 and 2 bills in the machine, if someone uses a 5, does it give back the change as two 1 bills or one 2 bill?2011-01-19
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    yes of course the point if in some point in time it out of those bills.2011-01-20
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    @Yakov, this is a random walk with step +3 (prob 2/3) and -2 (prob 1/3). This is easy to see: each buyer who pays exact change pays enough to give 2 change and have a 1-unit coin towards a second block of 2 change, so it suffices to count the total money in 1s and 2s present in the machine. What's not clear is precisely what you want to know about the walk. I think you can answer this most clearly by giving worked answers for n=1, n=2, n=3 according to your interpretation.2011-01-20

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