Question

2. Consider penalty shoot-outs in soccer that happen when the score is tied after the regulation playing time has expired (Let's call this state "tie"). Suppose that there two teams (Team A and Team B), and rounds of one kick per team, until the tie is broken (that is, starting from a tied score, there is a round of one penalty kick per team. If Team A scores, but team B doesn't, then Team A wins the game. But if both teams score (or neither of them scores), then we're again at a tied score, and there's an additional round of one penalty kick per team, and so on). Team A takes the first kick in the first shoot-out. Suppose that both Team A and Team B could make a penalty kick with the probability of 0.8 The winning team gets $10, and the losing team pays $10. Is this game recursive? a. b. Draw the game tree. If there is a state that recurs, do not go beyond that state. What is Team A's probability of winning the game in the state of tie? c. What is the probability of winning the game for Team A given it doesn't score on its first penalty d. kick? What is the value of the game? e.

Answer 1

Ciuen Ihat a Yes his me lace is t rn TEANM-A O Loss TEAM-B en ta bo earn A and Team oold make a penaH kick thproba biltyof 0.8 pro e Is P Cuimning aftr t tie) tp winning aftler and te) P(uoinning after a"etie)+ 0.04 O.G 1-004

0.04 C0.1 Gtt__ Co.IG)--о.9G- 0.30 0.04167(-1G) |-779 (0-16)十 - 0.2845十0.00 GG 0.291a ablit f team A inni andnet Scdinn a enal cam YO in 1 kick is nr3 tf( making 요ties utile rot Scaing at hei and uyn nira after tlot) P70 o.04 X o.29 Hence , from Baye's theore m, mot Sciing penalt, on st kick the

a Penal arn c.04 x 0.89 0.011G 0.058 eam MIMD observed (c) Team uinning is ake Some as sede Tearn A beCouse of t Same probail leam A B usnning & B tknce, Te probabili is $10xo.45十4(-15) xo.vs+串a rme Value game SZno e value of g