A prisoner is trapped in a cell containing 3 doors

#53. A prisoner is trapped in a cell containing 3 doors. The first door leads to a tunnel that returns him to his cell after 2 days travel. The second leads to a tunnel that returns him to his cell after 4 days travel. The third door leads to freedom after 1 day of travel. If it is assumed that the prisoner will always select doors 1, 2 and 3 with respective probabilities 0.5, 0.3, and 0.2. what is the expected number of days until the prisoner reaches freedom? (Question is also included in attachment)

The number of winter storms in a good year is Poisson random variable

#65. The number of winter storms in a good year is Poisson random variable with mean 3, whereas the number in a bad year is a Poisson random variable with mean 5. If next year will be a good year with probability 0.4 or a bad year with probability 0.6, find the expected value and variance of the number of storms that will occur. (Question is also included in attachment)

Find the joint mass function of X and Y

#39. Choose a number X at random from the set of numbers {1, 2, 3, 4, 5}. Now choose a number at random from the subset no larger than X, that is, from {1,…, X}. Call this second number Y. (a) Find the joint mass function of X and Y. (b) Find the conditional mass function of X given that Y = i. Do it for i = 1, 2, 3, 4, 5. (c) Are X and Y independent? Why? (Question is also included in attachment)

Two dice are rolled

#40. Two dice are rolled. Let X and Y denote, respectively, the largest and smallest values obtained. Compute the conditional mass function of Y given X = i, i = 1, 2, 3, 4, 5, 6. Are X and Y independent? Why? (Question is also included in attachment)

An urn has m black balls

#14. An urn has m black balls. At each stage a black ball is removed and a new ball, that is black with probability p and white with probability 1 - p, is put in its place. Find the expected number of stages needed until there are no more black balls in the urn. (Question is also included in attachment)