Complete questions (a) - (e) for the "Higher or Lower" exercise. If illustrations are necessary to complete the work, please include.

5. Higher or Lower Two contestants are playing a game called Higher or Lower. A prize is hidden inside one of several boxes placed in a row. The boxes are numbered from left to right 1, 2, 3,.... All boxes are equally likely to contain the prize. The players take turns to guess where the prize is hidden. If a guess is correct, then the host says 'correct and that player wins. If a guess is incorrect, the host, who knows where the prize is hidden, then says higher' or 'lower to faithfully indicate in which direction the prize box is located. Both players always follow the host's directions. For example, when there are 10 boxes numbered 1 to 10, with the prize in box 7, the game could progress like this: Player 1:3 Host: Higher Player 2:5 Host: Higher Player 1 : 8 Host: Lower Player 2:7 Host: Correct (a) A game has three boxes numbered 1, 2, 3. Show that if Player 1 chooses box 2 on the first turn, her chance of winning is 1/3, whereas if she chooses box 1 her chance of winning is 2/3. (b) A game has boxes numbered 1 to 4. Show that no matter which box Player 1 chooses on the first turn, Player 2 has a strategy for ensuring his chance of winning is 1/2. (c) A game has boxes numbered 1 to 9. Show that no matter which boxes Player 2 chooses, Player 1 has a strategy which involves choosing box 5 at the first turn and ensures her chance of winning is more than 1/2. At each turn of a game, a box with the smallest or largest number of those remaining is called an end box. (2) A game has an even number of boxes numbered 1 to n. If at each turn the player chooses an end box, show that each player's chance of winning is 1/2. () A game has an odd number of boxes numbered 1 to n. If at each turn the player chooses an end box, show that Player 1's chance of winning is 1/2(1+1)