What Is the Probability That A Survives the Duel?

[quasar987]

I find the questions of this week are more difficult than usual. I am in dire need of assistance!

A and B fight in a duel. They pick up their guns and fire once at each other. A kills B with probability $p_A$ and B kills A with probability $p_B$. If no one is killed, they repeat the process. What is...

a) The probability that A does not die

Sol:

$\Omega$: Every possible outcome of the duel.

$\Omega = \left\{ (A), (B), (AB), (d,A), (d,B), (d,AB), (d,d,A),..., (d,d,d,d,...)\right\}$

where A means "A wins", B measn "B wins", AB means "A and B both die" and "d" means a draw.

E: A does not die.

$E_i$: The duel lasts i rounds and end up with the death of B and the survival of A.

O: Nobody ever dies, i.e. it is the case of perpetual draws.

$$E=\bigcup_{i=1}^{\infty}E_i \cup O$$

Since these are all disjoint sets, P(E) is the sum of the probabilities, and we have

$$P(E_i) = [(1-p_A)(1-p_B)]^{i-1}p_A(1-p_B) \equiv r^{i-1}p_A(1-p_B)$$

(I made the hypothesis of independence btw the shots, i.e. P(A and B miss) = P(A misses)P(B misses))

$$P(O)=\lim_{i\rightarrow \infty}[(1-p_A)(1-p_B)]^{i-1}=0$$

$$\therefore P(E) = p_A(1-p_B)\sum_{i=0}^{\infty}r^i= \frac{p_A(1-p_B)}{1-r}$$