Three characters, A, B, and C, armed with guns, suddenly meet at the corner of a Washington, D. street, whereupon they naturally start shooting at...

1. Three characters, A, B, and C, armed with guns, 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. The probability of a hit for A, B, and C are α, β, and γ, respectively. A is the most hated, and therefore, as long as he is alive, B and C ignore each other and shoot at A. For historical reasons not developed here, A cannot stand B, and therefore he 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. What are the survival probabilities of A, B, and C, respectively?

Note: You may use Markov Chain.