I read a funny question about probability in a magazine. Here is the problem. A and B are to fight pistol duel. They are to fire at their choice of target in succession in the order A and B cyclically until only man is left unhit. A's chance of hitting B is 0.3, B's is 0.5. What's the probability for A to win finally? Here is the solution [tex]0.5*0.3 + 0.5^2*0.7*0.3 + 0.5^3*0.7^2*0.3 + \cdots [/tex] I am trying to understand this expression. A goes first. For the first term, A have 0.3 to shot B and 0.5 of not being killed. So if A can't kill B and not being killed in the first round. Then A is going to shot B again, so there exists the second term. But how do you understand this term? Why there is only one 0.3 and two 0.5? What does the 0.7 represent?