Miguel Mustang

Fix a positive integer $n$. Miguel is going to his local Mustang store, and the dealership offers him $2$ choices of motors. The first motor has $2n+1$ independent components, while the other motor has $2n-1$ independent components. Each of the components are functional with probability $0 < p < 1$, and both motors need at least half of their components to be functional for the motor itself to be operational. Miguel wants to maximize the probability his Mustang works, so he chooses the motor with $2n+1$ components whenever $p > p_n$, where $p_n$ is some threshold as a function of $n$. Find $p_{10}$.