Answer
The probability is 0.5. Yes.
Work Step by Step
As we see in figure 40.12b, the probability distribution function for a particle in a box that is in a stationary state is symmetric. The area under the right half of the curve is exactly the same as the area under the left half. The wavefunction is normalized, meaning the total area is 1, so each probability is exactly one-half.
This is true no matter what the value of n.
The answer makes sense because the entire situation is symmetric. The probability of finding the particle in the right half of the box must be the same as the probability of finding the particle in the left half of the box, and each probability is 0.5.