A particle executes an unrestricted simple random walk on the line with p=0.7 and q=0.3. Its position at time n is represented by the random variable Xn, n= 0, 1,...; Xo = 0. (a) Calculate the values of the following probabilities. (i) P(X5 = -2) (ii) P(X6 =2) (b) Calculate the probability that the particle returns to its starting point for the first time after 8 steps. (c) After the particle has been moving for some time, it is observed to be located at the point +3. (i) Calculate the probability that the particle will ever return to the origin from the point +3. (ii) Calculate the probability that starting from the point +3, the particle will visit the point +8 without first visiting the origin.

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Suppose that in a certain metropolitan area, 90% of all households have cable TV. Let x denote the number among four randomly selected households that have cable TV. Then x is a binomial random variable with   n = 4 and p = 0.90. (Round your answers to four decimal places.) (a) Calculate p(3) = P(x = 3). (b) Calculate p(4),the probability that all four selected households have cable TV. (c) Calculate P(x ≤ 3).

Let Y be a binomial random variable with n = 10 and p = 0.3. (a) P(3 < Y < 5) = P(3 ≤ Y < 5) = (b) P(3 < Y ≤ 5) = P(3 ≤ Y ≤ 5) =