You can see these things:

• If you shoot, then B will shoot, C won't
• If you don't shoot, B won't shoot, but C will.

So if you shoot your chances: $p_1(A)=0.7p(CA)+0.9(1-0.7)p(AB)+(1-0.7)(1-0.9)p_1(A)$ Where p(CA) is the probability to win a duel with C, if C is the first shooter, and p(AB) where you are the first shooter. $\begin{aligned}p(CA)&=0.7(1-0.5)+(1-0.7)(1-0.5)p(CA)\\ p(AB)&=0.7+(1-0.7)(1-0.9)p(AB)\end{aligned}$ So: $\begin{aligned}p_1(A)&=\cfrac{0.7\cfrac{0.7\cdot0.5}{1-0.3\cdot0.5}+0.9\cdot0.3\cfrac{0.7}{1-0.3\cdot0.1}}{1-0.3\cdot0.1}\\ &=\cfrac{0.7\cfrac{0.35}{1-0.15}+0.27\cfrac{0.7}{1-0.03}}{1-0.03}\\ &=\cfrac{\cfrac{0.245}{0.85}+\cfrac{0.189}{0.97}}{0.97}\\ &\approx50\%\end{aligned}$ And if you don't shoot, then B won't shoot too and C will shoot you, until he don't kill you, so your chances 0% in this case. $50\%>0\%\Rightarrow\boxed{50\%}$

Here is my C++ code:

Description: You can set these numbers: number_of_duels , max_precision , precision_A , precision_B , precision_C . This code doesn't use formulas because this code was written to show you the formulas I used are correct. With these settings the code will run for 10 seconds. With $10^8$ duels approxinetly for 15-20 minutes.

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#include <iostream>
#include <cstdlib>
#include <ctime>
#include <cmath>
using namespace std;
typedef long long int big;
big number_of_duels=10000000;

big sum_A;
big sum_B;
big sum_C;
/*Maximum precision is 100*/
int max_precision=100;

int precision_A=70;
int precision_B=90;
int precision_C=50;

int dont_shoot_A=0;
int dont_shoot_B=0;
int dont_shoot_C=0;

const int A=0;
const int B=1;
const int C=2;

void zeros(int x);
int shoot(int x);

int A_B_C();
int B_C_A();
int C_A_B();

int C_A();
int A_C();
int A_B();
int B_A();
int B_C();
int C_B();

int main()
{
    srand(time(0));
    int temp;
    for(dont_shoot_A=0;dont_shoot_A<2;dont_shoot_A++){
        for(dont_shoot_B=0;dont_shoot_B<2;dont_shoot_B++){
            for(dont_shoot_C=0;dont_shoot_C<2;dont_shoot_C++){
                if(dont_shoot_A==1&&dont_shoot_B==1&&dont_shoot_C==1) break;
                sum_A=0;
                sum_B=0;
                sum_C=0;
                for(int current_duel=0;current_duel<number_of_duels;current_duel++){
                    temp=A_B_C();
                    if(temp==A) sum_A++;
                    else if(temp==B)    sum_B++;
                    else if(temp==C)    sum_C++;
                    else cout << "error!!!!!";
                }
                cout << "\n\nA:" << dont_shoot_A << "\tB:" << dont_shoot_B << "\tC:" << dont_shoot_C;
                cout << "\n0.";
                zeros(A);
                cout << sum_A;
                cout << "\t0.";
                zeros(B);
                cout << sum_B;
                cout << "\t0.";
                zeros(C);
                cout << sum_C;
            }
        }
    }
    return 0;
}
void zeros(int x){
    int zeros_number_of_duels=log10(number_of_duels);
    int zeros_sum;
    if(x==A){
        if(sum_A)    zeros_sum=log10(sum_A);
        else return;
    }else if(x==B){
        if(sum_B)    zeros_sum=log10(sum_B);
        else return;
    }else if(x==C){
        if(sum_C)    zeros_sum=log10(sum_C);
        else return;
    }
    for(int y=1;y<zeros_number_of_duels-zeros_sum;y++)  cout << 0;
}
int shoot(int x){
    if(x==A)  return ((rand()%max_precision+1)>precision_A);
    if(x==B)  return ((rand()%max_precision+1)>precision_B);
    if(x==C)  return ((rand()%max_precision+1)>precision_C);
}
int A_B_C(){
    int missed=shoot(A);
    if(!dont_shoot_A)    return (missed)? B_C_A():C_A();
    else return B_C_A();
}
int B_C_A(){
    int missed=shoot(B);
    if(!dont_shoot_B)    return (missed)? C_A_B():A_B();
    else return C_A_B();
}
int C_A_B(){
    int missed=shoot(C);
    if(!dont_shoot_C)   return (missed)? A_B_C():B_C();
    else return A_B_C();
}
int C_A(){
    int missed=shoot(C);
    return (missed)? A_C():C;
}
int A_C(){
    int missed=shoot(A);
    return (missed)? C_A():A;
}
int A_B(){
    int missed=shoot(A);
    return (missed)? B_A():A;
}
int B_A(){
    int missed=shoot(B);
    return (missed)? A_B():B;
}
int B_C(){
    int missed=shoot(B);
    return (missed)? C_B():B;
}
int C_B(){
    int missed=shoot(C);
    return (missed)? B_C():C;
}

Output:

Without the first and second lines. max_precision=1000,000,000

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A:0 B:0 C:0
0.490251313 0.090364688 0.419383999

A:0 B:0 C:1
0.497812583 0.077167163 0.425020254

A:0 B:1 C:0
0.338967042 0.166930172 0.494102786

A:0 B:1 C:1
0.411526617 0.0 0.588473383

A:1 B:0 C:0
0.684252147 0.312983441 0.002764412

A:1 B:0 C:1
0.722186746 0.277813254 0.0

A:1 B:1 C:0
0.0 0.947433430 0.052566570