Three characters, A, B, and C, each armed with a gun, suddenly meet at the corner...

 

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Three characters, A, B, and C, each armed with a gun, suddenly meet at the corner of a Washington D.C. street, whereupon they naturally start shooting at one another. Each street-gang kid shoots every tenth second, as long as he is still alive. Each street-gang kid has a different initiative though: A is fastest in drawing his gun and starts shooting at ta = 0 seconds; B is less adept and starts shooting at tв = 1 seconds; finally, C is the slowest and starts shooting at to 2 seconds. Also, each street-gang kid is differently prepared: A and B each have two bullets each, whereas C has only one bullet. The probability of a successful hit for A, B, and C are a, 3, 7 € (0,1), respectively. A is the most hated, and therefore, as long as he is alive, B and Cignore each other and shoot at A. For historical reasons not developed here, A cannot stand B, and therefore shoots only at B while the latter is still alive. Lucky C is shot at if and only if he is in the presence of A alone or B alone. (a) Formulate this battle as a Markov chain. Be sure to: i. Clearly indicate the state space E, as well as which states are transient and which are absorbing. ii. Write down all transition probabilities of the Markov chain. iii. Draw a schematic representation of the Markov chain, clearly indicating all states and transition probabilities. (b) Give formulae for each of the survival probabilities of A, B, and C, respectively. These will be functions of a, 3, 7. Three characters, A, B, and C, each armed with a gun, suddenly meet at the corner of a Washington D.C. street, whereupon they naturally start shooting at one another. Each street-gang kid shoots every tenth second, as long as he is still alive. Each street-gang kid has a different initiative though: A is fastest in drawing his gun and starts shooting at ta = 0 seconds; B is less adept and starts shooting at tв = 1 seconds; finally, C is the slowest and starts shooting at to 2 seconds. Also, each street-gang kid is differently prepared: A and B each have two bullets each, whereas C has only one bullet. The probability of a successful hit for A, B, and C are a, 3, 7 € (0,1), respectively. A is the most hated, and therefore, as long as he is alive, B and Cignore each other and shoot at A. For historical reasons not developed here, A cannot stand B, and therefore shoots only at B while the latter is still alive. Lucky C is shot at if and only if he is in the presence of A alone or B alone. (a) Formulate this battle as a Markov chain. Be sure to: i. Clearly indicate the state space E, as well as which states are transient and which are absorbing. ii. Write down all transition probabilities of the Markov chain. iii. Draw a schematic representation of the Markov chain, clearly indicating all states and transition probabilities. (b) Give formulae for each of the survival probabilities of A, B, and C, respectively. These will be functions of a, 3, 7.

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