Solve problem of Exercise 28 for 2ⁿ, n > 1, boxes. Does the above probability increase or decrease as n is increasing? Find the limit as n→∞.

Exercise 28

There are four closed boxes, one of which contains a prize and the others are empty. A player can consecutively check boxes until she/he finds the prize, and in this case, she/he pays $1 for each check (including the last box if the prize is there). The player may also divide the boxes into two equal parts and ask the host of the game in which group the box with the prize is. After the answer, the player asks the host which box among the two contains the prize.
Each such a question also costs $1. Let X and Y be the total payment for the first and second strategy, respectively. Which strategy is better on the average? Compute E{X} and E{Y}.
With what probability will the strategy that is worse on the average, lead to a better result?