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MATH3901 Probability and stochastic processes (1 Viewer)

Superbox

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A slot machine works on inserting $1 coin. If the player wins,
the coin is returned with an additional $1 coin, otherwise the original coin is
lost. The probability of winning is 1/2 unless the previous play has resulted
in a win, in which case the probability is p < 1/2. If the cost of maintaining
the machine averages $c per play (with c < 1/3), give conditions on the value
of p that the owner of the machine must arrange in order to make a profit in
the long run.


Not sure how to start this. Is this a markov chain (gambler's ruin)
 

InteGrand

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A slot machine works on inserting $1 coin. If the player wins,
the coin is returned with an additional $1 coin, otherwise the original coin is
lost. The probability of winning is 1/2 unless the previous play has resulted
in a win, in which case the probability is p < 1/2. If the cost of maintaining
the machine averages $c per play (with c < 1/3), give conditions on the value
of p that the owner of the machine must arrange in order to make a profit in
the long run.


Not sure how to start this. Is this a markov chain (gambler's ruin)




 
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Superbox

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A flea hops on the vertices A, B, and C of a triangle. Each hop takes it from one vertex to the next and the times between sucessive hops are independent random variables, each with an exponential distribution with mean 1/λ. Each hop is equally likely to be in the clockwise direction or in the anticlockwise direction. Find the probability that the flea is at vertex A at a given time t>0, starting from A at time t=0.

(Hint: Write the Kolmogorov's backward equations and solve for the transition probability function of interest. The solution of y′(x)=a+by(x) is y(x)=c*e^(bx)−a/b, for some constant $c.)

Have another question. I wrote out the transition instanteuous rate matrix and there are 9 back ward equation to solve. Not sure how to solve these equations using the hint.
 

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