Problem #PRU-65325

Problem

Hercules meets the three-headed snake Hydra of Lerna. Every minute, Hercules chops off one head of the snake. Let $x$ be the survivability of the snake ($x>0$). The probability $p_s$ of the fact that in the place of the severed head will grow s new heads $(s=0,1,2)$ is equal to $\frac{x^s}{1+x+x^2}$.

During the first 10 minutes of the battle, Hercules recorded how many heads grew in place of each chopped off one. The following vector was obtained: $K=(1,2,2,1,0,2,1,0,1,2)$. Find the value of the survivability of the snake, under which the probability of the vector $K$ is greatest.