A man plays independent repetitions of the following game: at each repetition, he throws a dart onto a circular target. Suppose that the distance D between the impact point of the dart and the center of the target has a U[0,30] distribution. If D ≤ 5, the player wins $1; if 5 Xn be the fortune of the player after n repetitions. Then the stochastic process {Xn, n = 0 , 1 , . . . } is a Markov chain.
(a) Find the one-step transition probability matrix of the chain.
Suppose now that the man never stops playing, so that the state space of the Markov chain is the set {0, ± 1 , ± 2 , . . . } Suppose also that the duration T (in seconds) of a repetition of the game has an exponential distribution with
mean 30. Then the stochastic process {N(t),t ≥ 0}, where N(t) denotes the number of repetitions completed in the interval [0, t], is a Poisson process with rate λ = 2 per minute.
(c) Calculate the probability that the player will have completed at least three repetitions in less than two minutes (from the initial time).
(d) Calculate (approximately) the probability P[N(25) ≤ 50].