Independent trials, each resulting in a success with probability p, are performed until k consecutive successful trials have occurred. let x be the total number of successes in these trial, and let pn = p(x = n). (a) find pk. (b) derive a recursive equation for the pn, n ≥ k, by imagining that the trials continue forever and conditioning on the time of the first failure. (c) verify your answer in part (a) by solving the recursion for pk. (d) when p = .6 , k = 3, find p8 .

Luke started a weight-loss program. the first week, he lost x pounds. the second week, he lost pounds less than times the pounds he lost the first week. the third week, he lost 1 pound more than of the pounds he lost the first week. liam started a weight-loss program when luke did. the first week, he lost 1 pound less than times the pounds luke lost the first week. the second week, he lost 4 pounds less than times the pounds luke lost the first week. the third week, he lost pound more than times the pounds luke lost the first week.