5. The average rank for the entire deck of 52 cards is 8. a. Verify this claim. Proof: Equation 3.1: = (x) = = eg(x) 1+eg(x) ... Equation 3.2: g(x) = 0 + 1x1 + 22 + + pxp ... Equation 3.3: In () = B + B 1x1 + Bx2 + + pxp 1- Equation 3.6: In (1) = 6.8338+0.8579 (Average Rank) Equation 3.7: exp (6.8338+0.8579 (Average Rank)) = 1+exp (=-6.8338+0.8579(Average Rank)) b. If your initial average rank is 8, what is the initial average rank of your opponent (in a two-player game)? c. What, intuitively should be the estimated chance of winning if your initial average rank is 8? In (7) = 6.8338 + 0.8579 (Average Rank) In - =-6.8338+0.8579(8) In = 0.0294 2.94% d. What does equation (3.7) predict for your chance of winning if your initial average rank is 8? d. What does equation (3.7) predict for your chance of winning if your initial average rank is 8? Equation 3.7 : exp (6.8338+0.8579(Average Rank)) = 1+exp (=-6.8338+0.8579(Average Rank)) = exp (-6.8338+0.8579(8)) 1+exp (=-6.8338+0.8579(8)) exp (0.0294) = 1+exp (0.0294) 1.029836 = 1+1.029836 1.029836 == 2.029836 =0.5073494 0.507 What this value tells about your chances of winning is that you have a 0.507 or 50.7% chance of winning.