3. A video game has 40 stages, each getting progressively harder to complete. When a stage is successfully completed, the player moves to the next level. If a stage is not completed successfully, the player must repeat the stage. The probabilities of success at any stage stays the same regardless of how many times the player has played that stage. At stage k, the probability of successfully moving to the next stage is 1/k. When you get to stage 40, the game is over. (a) Starting at stage 1, what is the expected number of total stages played before finishing the game (that is, getting to stage 40)? For example, if stages 1,2,3,4,4,4,5,5,6 are played in that order, it counts as 9 stages played. (b) In a variation of the game, a player gets k points for successfully completing the kth stage but loses k points for failing and then having to repeat any stage. What is the expected number of points a player will have at the end of the game? When you got to stage 40 you got 392 points for getting there but the game is over. (Hint: Let Wk be the expected number of points you will subsequently earn every time you play stage k. What is a formula relating the various Wk to each other?)