30 / 185 * 11 = 1.7
1.7 ~= 0, so the sample size is too small to support the headline.
Drawing 100,000 samples from these distributions (`betarnd` function in matlab) gives an 87.8% sampled likelihood that the intervention reduces the conditional risk (intervention(i) < control(i)), and a 64% sampled likelihood that it reduces the risk by at least half (intervention(i) < control(i) / 2).
This is suggestive (but not yet "clearly convincing") evidence that the vaccine reduces the risk of severe covid conditional on being infected in the first place, and that comes on top of the obviously compelling evidence that the vaccine reduces the baseline risk of infection.
This conclusion makes intuitive sense since the vaccine is intended to produce an immune response. A patient who has a moderate response to the vaccine itself may not neutralize the infection before developing symptoms but would still have a primed immune system to control the disease before it becomes severe. (That is: the response to the vaccine intuitively falls on a range, rather than being "all or nothing").
Thanks for running through the math; I’m guessing we came to the same conclusion about the strength of the evidence.