Question

In a game of repeated die rolls, a player is allowed to roll a standard die up to ​n​ times, where ​n​ is determined prior to the start of the game. On any roll except the last, the player may choose to either keep that roll as their final score, or continue rolling in hopes of a higher roll later on. If the player rolls all ​n times, then after the ​n​th roll, the player must keep that roll as their final score. A player always acts to maximize their expected final score. Finally, let V​n​ denote the final score in a game with a max of ​n​ rolls allowed. a) Compute E[V​2]​ with justification. b) Compute E[V​3]​ with justification. c) Find the smallest​ n ​such thatE[V​n]​ ≥5. d) Find the smallest ​n ​such that E[V​n]​ ≥5.999. Hint:try finding a closed-form expression for E[V​n]​ .

In a game of repeated die rolls, a player is allowed to roll a standard die up to n times, where n is determined prior to the start of the game. On any roll except the last, the player may choose to either keep that roll as their final score, or continue rolling in hopes of a higher roll later on. If the player rolls all n times, then after the nth roll, the player must keep that roll as their final score. A player always acts to maximize their expected final score. Finally, let V, denote the final score in a game with a max of n rolls allowed a) Compute E[V] with justification. b) Compute E[V,3] with justification. c) Find the smallest n such that E[V] 2 5 d) Find the smallest n such that E[V] 2 5.999. Hint: try finding a closed-form expression for E[V

Answer 1

Solution: from the given data, we uill generate this for Va Now 6. (only tilln) a the playen wl play until he gets Probability n- 우용+우 thtrial $kial 1-5/6 우음x 을x응 Out Come of 5 X n-1

The Same for 4, 3, 2,1 NOw E[VJ- 6(1-1)) +(5413 +2+)) 6- . E [V] 6-3 Now, 141 2S EEV.J. 6-3( 6-3 3.9167 E J- 6-3 L26389 b -> 4.26389 (C The Smallest n such that E 6-3 >5 1Z3 3 Hence the Smallest n- 7 (P ElvnJ 5.999 6-3(96)2 5.499 n 3000 nN43.91 So, the Smallest 44