Probability that the fox will survive the hunt **Problem** A hunter sees a fox $40$ meters away and shoots at him. If the fox survives the shot, he runs away $10$ m/s and the hunter continues to shoot every $2$ seconds until she kills him or the fox vanishes. The probability that the hunter will shoot the fox is $10000\cdot x^{-3}$ where $x\ge 40$ is the distance between the hunter and the fox. If the fox gets shot, the probability that he will die is $1/5$, independent of how many times he got shot. What is the probability that the fox will survive? **My solution** Let $B$ be the event that the fox dies and $A_i$ be the event that the fox dies at the $i$th shot. Since $A_i$ are mutually exclusive we have $P(B)=\sum_{i=0}^{\infty}P(B|A_i)P(A_i)=10000\cdot 4/5\cdot\sum_{i=0}^{\infty}1/(40+20i)^3\approx 0.2$. Therefore the probability that the fox survives is $1-0.2=0.8$. I'm not really sure about this though. It is suspiciously likely that the fox survives.

The conditional probability of the fox surviving shot #$i$, given that it survives up to that point, is $1 - 10000/(5 (40 + 20 i)^3) = 1 - 1/(4 (2+i)^3)$, $i=0, 1, \ldots$. Thus the probability of eventual survival is $$ \prod_{i=0}^\infty \left(1 - \frac{1}{4 (2 + i)^3}\right) $$ Maple says this is $$ {\frac {-16}{3\; \Gamma \left( -\sqrt [3]{2}/2 \right) \Gamma \left( \left(1-i\sqrt{3}\right) \sqrt[3]{2}/4 \right) \Gamma \left( \left( 1+i\sqrt {3} \right) \sqrt [3]{2}/4 \right) }} \approx 0.950215097 $$