The mean free path ($\lambda$) of a gas molecule is inversely proportional to the number density ($n$) of the gas molecules, given by the formula $\lambda = \frac{1}{\sqrt{2}\pi n d^2}$. The pressure ($P$) of an ideal gas is directly proportional to the number density at a constant temperature ($P = nkT$). Combining these relationships, the mean free path is inversely proportional to the pressure ($\lambda \propto \frac{1}{P}$). Therefore, if the mean free path is doubled, the pressure must be halved to maintain the relationship.