5. The average rank for the entire deck of 52 cards is 8. a. Verify this claim. b. If your initial average rank is 8, what is the initial average rank of your opponent (in a two-player game)? A standard 52 card deck comprises of 13 ranks in each of the four suits: clubs, diamonds, hearts, and spades. Each suit includes three court cards: King, Queen, and Jack The ranks of these range from 2-10 for the four suits. 11 for Jack, 12 for Queen, 13 for King, and 14 for Ace. Equation 3.1: = (x) eg(x) = 1+eg(x) Equation 3.2: g(x) = 0 + 1x1 + 22 + + pxp Equation 3.3: In (7 Bo+B1x1 + Bx2 + = . . . + Bpxp Equation 3.6: In (^^) = 6.8338 + 0.8579(Average Rank) Equation 3.7: = exp (6.8338+0.8579(Average Rank)) 1+exp (=6.8338+0.8579(Average Rank)) One way to do this using the data is to run the war.txt file and do mean(war$RankAvg) to get 8.015002. In (117) = 6.8338+0.8579 (Average Rank) In (1777) = 6.8338+0.8579() In ( In = 6.8338+0.8579() = In the children's card game of 'War' between two players, each player is randomly given half of the deck of 52 cards. Each player has their cards in a pile face down. Players simultaneously turn over the card at the top of their pile and display it in the playing area between the players so that both players can see. If one player has a card with a higher rank, then she takes her opponent's card and her own, and places them face down at the bottom of her pile. What makes the game interesting is that the two cards drawn may have the same rank (then a War' ensues). With cards of the same rank (at least in one variation of the game) each player places the next card from the top of their pile face down on their tied-rank card, followed by another card displayed face up. If one player then has a higher rank, then she takes all cards and places them face down at the bottom of her pile. If tied-rank cards occur again, the process of placing a facedown card followed by a face up card is repeated until one player has a face up card of a higher rank. The game ends when one player has all 52 cards. In the above, rank is simply the denomination for cards showing the values 2-10. Rank is 11 for a Jack, 12 for a Queen, 13 for a King, and 14 for an Ace. The file War.txt contains the results of 1,000 simulated games of War. In particular, each row in this text file shows the results of a single game from the perspective of one of the players, call him player A. The variables in this text file are described in Table 3.3. The number of Aces was deemed important as these are the most powerful (have the highest rank). Note, however, from the game rules, that Aces are not guarranted to stay with a single player during a game. Variable Name Win RankAvg Battles Turns Acel Ace2 Ace3 Ace4 Aces Table 3.3: War.txt Variables Definition 1 if player A wins, 0 if player A loses Average initial rank for player A Total number of times the players compare card values A turn is completed whenever a player takes cards 1 if player A had one ace at game start, 0 otherwise 1 if player A had two aces at game start, 0 otherwise 1 if player A had three aces at game start, 0 otherwise 1 if player A had four aces at game start, 0 otherwise Number of Aces player A had at game start