We can not reliably say whether exposure N (mSv) will cause cancer in person P - we can only predict it based on previous observations.
Predicting the effect from radiation exposure is based on long-term epidemiological study data such as the Japan Life Span Study [1-3]. These compare the disease rates in large populations to neighbouring/control populations where radiation exposure was at natural levels.
These studies form the basis of a statistical reference when establishing the likelihood of developing an illness due to radiation exposure. They suggest that there is a ‘statistically significant increase of the risk of fatal cancer starting at the range of 50–100 mSv, possibly already at 10–50 mSv’ .
TFA: "Residents of Namie town and Iitate village, two areas that were not evacuated until months after the accident, received 10–50mSv"
Deterministic effects (i.e. observed reliably above a certain dose threshold) of exposure are seen above 100mSv .
TFA: "146 employees and 21 contractors received a dose of more than 100 millisieverts (mSv), the level at which there is an acknowledged slight increase in cancer risk. Six workers received more than the 250mSv allowed by Japanese law for front-line emergency workers, and two operators in the control rooms for reactor units 3 and 4 received doses above 600mSv".
Through previous observations of population exposures to radiation at similar levels, it is statistically likely that this accident will result in an increase in cancer incidence among this population.
 Preston, D.L., et al., Cancer Incidence in Atomic Bomb Survivors. Part III: Leukemia, Lymphoma and Multiple Myeloma, 1950-1987 Radiation Research, 1994. 137 (2 (Suppliment)): p. S68 - S97.
 Preston, D.L., et al., Solid Cancer Incidence in Atomic Bomb Survivors: 1958–1998. Radiation Research, 2007. 168(1): p. 1-64.
 Land, C.E., Studies of Cancer and Radiation Dose Among Atomic Bomb Survivors. The Example of Breast Cancer. JAMA, 1995. 274(5): p. 402 - 407.
 Vock, P., CT Dose Reduction in Children. European Radiology, 2005. 15: p. 2330-2340.