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NCI declined to provide estimates of how many cancers and deaths might result from this population exposure. I made the estimates given here (and on various test series pages accessible from this one) from material on the NCI site in the following way.

The incidence of thyroid cancer increases 7.7 fold for each 100 rad exposure. The base lifetime risk of contracting thyroid cancer is 0.47% (a likelihood of 0.0047) averaged over both men and women. 100 rads of exposure thus increases the risk to 3.62% (0.0362), which is an additional incidence of 3.15% (0.0315) due to the radiation. Making the standard assumption of linearity in biological response to radiation dose, the 380 million rad population exposure thus would result in 120,000 additional cases of thyroid cancer. The mortality rate of thyroid cancer is given as 5%, so about 6,000 deaths would result.

Now the 7.7 fold risk enhancement figure is actually for exposures to young children, so the estimate given here is probably something of an overestimate. However two factors should be kept in mind. First, these radiation exposures occurred at the height of the Baby Boom (the peak year was 1957), when the proportion of the population consisting of young children was extraordinarily large. Second, the principal route for radiation exposure was drinking contaminated milk. Children drink a disproportionate amount of milk and thus accumulated a large part of the total population radiation exposure.