Mass ratio $q$ of a contact binary star evolves due to mass transfer, magnetic braking, and thermal relaxation oscillations to small values until it crosses a critical threshold $q_\text{min}$. When that happens, the binary undergoes the tidal Darwin instability leading to a rapid coalescence of the components and observable brightening of the system. So far, distribution of $q$ and $q_\text{min}$ have not been measured on a sufficiently large population of contact binary stars, because the determination of $q$ for a single contact binary usually requires spectroscopy. But as was shown previously, it is possible to infer the mass-ratio distribution of the entire population of contact binaries from the observed distribution of light curve amplitudes. Employing Bayesian inference, we obtain a sample of contact-binary candidates from the Kepler Eclipsing Binary Catalog combined with Gaia data and estimates of effective temperature. Each candidate is assigned a probability of being a contact binary of either late or early type. Overall, our sample includes about 300 late-type and 200 early-type contact binary candidates. We model the amplitude distribution assuming that mass ratios are described by a power law with exponent $b$ and a cut off at $q_\text{min}$. We find $q_\text{min}=0.087^{+0.024}_{-0.015}$ for late-type contact binaries with periods longer than 0.3 days. For late-type binaries with shorter periods, we find $q_\text{min}=0.246^{+0.029}_{-0.046}$, but the sample is small. For early type contact binary stars with periods shorter than 1 day, we obtain $q_\text{min}=0.030^{+0.018}_{-0.022}$. These results indicate a dependence of $q_\mathrm{min}$ on the structure of the components and are broadly compatible with previous observations and theoretical predictions. We do not find any clear trends in $b$. Our method can be easily extended to large samples of contact binaries from TESS and other space-based surveys.