Due: Wednesday, February 9 at 10:59pm (submit via Gradescope). Policy: Can be solved in groups (acknowledge collaborators) but must be written up individually Submission: Your submission should be a PDF that matches this template. Each page of the PDF should align with the corresponding page of the template (page 1 has name/collaborators, question 1 begins on page 2, etc.). Do not reorder, split, combine, or add extra pages. The intention is that you print out the template, write on the page in pen/pencil, and then scan or take pictures of the pages to make your submission. You may also fill out this template digitally (e.g. using a tablet.) First name Last name SID Collaborators For sta↵ use only: Q1. Expectimax Yahtzee /40 Total /40 1 Muskaan Goyal 3032666782 MIA This study source was downloaded by 100000840858457 from CourseHero.com on 07-06-2022 03:01:12 GMT -05:00 https://www.coursehero.com/file/130827698/cs-188-spring-2022-written-homework-2-2pdf/ Q1. [40 pts] Expectimax Yahtzee Consider a simplified version of the game Yahtzee. In this game, we have 3 regular tetrahedral dice with 4 sides each (numbered 1-4) and the game begins by rolling all 3 dice. At this point, a player can make a decision: pick one of the 3 dice to reroll, or don’t reroll anything. Then, points are assigned as follows: • A reward of 10 points is given for two-of-a-kind (for example, 4-4). • A reward of 15 is given to three-of-a-kind (for example, 4-4-4). • A reward of 7 points is given for rolling a series (1-2-3 or 2-3-4). • Otherwise (or if the sum is higher than the special reward), the score is equal to the sum of all 3 dice. (a) We will formulate this problem as an expectimax tree. (i) [3 pts] The resulting tree for the problem is drawn below. Given a specific initial roll, the branching factor (of the player’s decision) from the root node is . The branching factor at the chance nodes is . What do those chance nodes represent? (There are multiple solutions, you only need to write down one solution) • Chance node 1: • Chance node 2: • Chance node 3: 2 4 4 neroll dice I renew dice 2 renew dice 3 max special reward I 2 3 4 I 2 3 4 I 2 3 Y This study source was downloaded by 100000840858457 from CourseHero.com on 07-06-2022 03:01:12 GMT -05:00 https://www.coursehero.com/file/130827698/cs-188-spring-2022-written-homework-2-2pdf/ (ii) [7 pts] Given a starting roll (1,2,4) (corresponding to the outcomes of die rolls 1, 2, and 3 respectively), what move should you take? Fill in the values of the expectimax tree below to justify your answer. 3 no recall poor died puog.IT Refs 9 as 7 ( 1,214) Suma 7 7 10 9 10 10 7 8 10 10 10 7 7 Reroddiu reward Rerotldrcel : Reynard ① ( 41,4) 1 : ① 41214 ) f | ② ( " " "I 7 ② (2,214 ) 1° ③ ( 1,314) ⑨ ( 114,4) 10 ③ 1312,4) 9 ④ (4/44) 10 " "%→""""" As revolting any dice gives better rewards, RuoUDñYpqand so we should choose sending over ① ( 112,1) 10 not no - revolt . ② ( 1,2 , 2) 10 ③ 11 , 2,3) ⑨ 1112,4) / 7 Average → 3414--85 / This study source was downloaded by 100000840858457 from CourseHero.com on 07-06-2022 03:01:12 GMT -05:00 https://www.coursehero.com/file/130827698/cs-188-spring-2022-written-homework-2-2pdf/ Now suppose the human player does not understand how to play the game, and as a result, they choose any action with uniform probability, regardless of the initial roll. Moreover, we assume that the human’s choice will be carried out by a ”somewhat helpful” robot called Albertbot: given a configuration of dice and the desired action from the human, this robot either actually implements the human’s action (with probability 1