Consider this variation of the single-player game Minesweeper. An area of land is divided into a 100 sections by a 10 by 10 grid. Each section of land has a landmine buried in it with probability 10, independent from section to section. You will choose a single number k, the number of sections you will test for mines. Since the mine placement is randomized, we may assume that you test consecutive indexed sections 1 through k: from the the Whole set of sections numbered 1 through 100. If all k sections you test are free of mines, you end the game With It points. If one of the sections has a mine you end up with zero points. (a) (b) (C) (01) Take It to be a number, so that all the randomness of the \"experiment\" here is in the placement of the mines. Describe the outcome space 9. Let X k be the random variable equal to the number of landmines in sections 1 through It. What is the PMF of Xk? What is P(X;C = 0)? What is E[Xk]? Let Y], be the value of the score in points if you play the strategy of revealing sections 1 through 16. Thus y_ 1.: ika=0, '9' 0 ika>0. What is E[Yk], in terms of k and p? What value of k maximizes E[Yk] when p = 0.05? Hint: maximizing the log() of a function is equivalent to maximizing the original function, since logo is monotone increasing. Take the logo (natural log) of your expression from part c, and then take the derivative of the resulting expression with respect to It. Find What value of It makes this derivative 0, and call that value 16*. If [9* is not an integer, test your expression from part c at oorUc'\") and ceil(lc*). It might be that both of these integers are maximizing values