A gnome hunter needs to find two gnomes hidden in the room. Assume that the probability to find a gnome in a short time interval of length h is equal to 2h + o(h). Independently of that, a gnome found by the hunter can escape and hide at a new location in the room: the probability that this occurs in a time interval of length h is equal to h + o(h). However, once the hunter holds both gnomes, he ties them so that they can no longer escape. (a) Construct a continuous-time Markov chain describing this process. Write down the infinitesimal generator and the initial distribution. (b) What is the distribution of the time until the hunter manages to find the first gnome? (c) Let pn(t) be the probability that the hunter is holding n gnomes at time t (n = 0, 1, 2). Write down a system of differential equations satisfied by these three functions. (d) What is the probability that the hunter will find both gnomes by time t?