# MATH3901 Probability and stochastic processes (1 Viewer)

#### Superbox

##### New Member
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 ##### Well-Known Member 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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• leehuan and BLIT2014

#### Superbox

##### New Member
Thanks for the details response. Is the answer $p < \frac{1 -3c}{2(1 - c)}$ (no idea how to make latex work.

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#### Superbox

##### New Member
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.