# Sir Lancelot and Sir Galahad are doing a shoot out, in which they try to shoot each other while shooting in the same time at each other. The probability of Sir Lancelot to hit Sir Galahad is 0.5 and the probability of Sir Galahad to hit Sir Lancelot is 0.25. All the shots are independent.

The probability of winning in a shootout, using a geometric random variable
Sir Lancelot and Sir Galahad are doing a shoot out, in which they try to shoot each other while shooting in the same time at each other. The probability of Sir Lancelot to hit Sir Galahad is 0.5 and the probability of Sir Galahad to hit Sir Lancelot is 0.25. All the shots are independent.
A. What is the probability that the shoot out will end in the n's round ?
B. If it is known that after k rounds the shoot out did not end, what is the chance that is will end within two rounds ?
C. What is the chance of Sir Lancelot to win ?
D. What is the chance of Sir Galahad to win ?
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Kody Gates
Step 1
By L wins, I assume what is meant is that L survives and G does not. Let p be the probability L wins.
Step 2
We condition on what happens in the first round. L wins if either (i) he hits G, and G misses or if (ii) they both miss, but ultimately L wins. We therefore obtain $p=\left(0.5\right)\left(0.75\right)+\left(0.5\right)\left(0.75\right)p,$ since in effect if they both miss the party begins all over again. Solve this linear equation for p.
The Galahad wins probability is calculated in the same way
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ghulamu51
Step 1
L wins in the following pairwise disjoint ways:
(i) L makes his first shot, and G misses, probability (0.5)(0.75).
(ii) They both miss their first shots, but L nails the second shot, and G misses, probability (0.5)(0.75)(0.5)(0.75).
(iii) They both miss their first two shots, but then L hits on the third, and G misses, probability [(0.5)(0.75)]2(0.5)(0.75).
(iv) And so on.
So the probability L wins is $\left(0.5\right)\left(0.75\right)\left[1+r+{r}^{2}+{r}^{3}+\cdots \right],$ where $r=\left(0.5\right)\left(0.75\right)$. If we sum the infinite geometric series, we get $\frac{\left(0.5\right)\left(0.75\right)}{1-\left(0.5\right)\left(0.75\right)}$