From the page I linked:

The correct answers to hospitalization risk can be calculated using data from the Department of Health and Human Services (via HealthData.gov) and the U.S. Centers for Disease Control and Prevention (CDC). One needs only the following figures: 1) the population of vaccinated and unvaccinated people 2) total hospitalizations resulting from COVID-19 3) hospitalizations of vaccinated people. We used data through August 9, 2021, one week before the survey was fielded. At that time, total hospitalizations from COVID-19 were estimated to be 2.6 million, with 7,608 vaccinations found among vaccinated people. The size of the vaccinated and unvaccinated populations was nearly equal on August 9 (with 168 million vaccinated and 163 million unvaccinated).

A simplistic analysis of these numbers would yield hospitalization rates of 0.005% for the vaccinated population (1 case in 22,118) and 1.6% for the unvaccinated population (1 case in 62), but those numbers exaggerate the benefits of the vaccine because the unvaccinated population confronted many more days of risk, since vaccination was gradually rolled out starting in December of 2020. For that reason, we take the average population totals over the relevant periods for each population (March 1, 2020-Aug. 9, 2021 for the unvaccinated population and Dec. 15, 2020,-Aug. 9, 2021, for the vaccinated population). The adjusted population of vaccinated people comes to 83 million and 295 million for the vaccinated population, since the entire U.S. population was unvaccinated -- except a small number of participants in clinical trials --up until December of 2020.

Using these adjusted figures, we calculate that the hospitalization rate for the vaccinated population is 0.01% (or 1 in 10,914), and the rate for unvaccinated adults is 0.89% (or 1 case in 112 people). In both cases, therefore, the correct answer is less than one percent, but the implied efficacy rate of vaccination is 99% at preventing hospitalizations. This is calculated as the hospitalization rate for the unvaccinated minus the hospitalization rate for the vaccinated, divided by the unvaccinated rate. In other words, it is the percentage decrease in hospitalization risk. This high rate of protection -- even against Delta -- is consistent with a recent article published in the Lancet, which reviewed large-scale empirical data from the United States and around the world.

Some may argue that patients may have been hospitalized as a result of COVID but not diagnosed as such. We think this is highly unlikely to result in significant downward bias in the rates of hospitalization risk since testing at hospitals became widespread after only a few weeks at the start of the pandemic, and the vast majority of hospitalizations would have occurred since May of 2020, given data on deaths, which are more comprehensively documented. Nonetheless, using various modeling assumptions, CDC epidemiologists estimate that the actual number of hospitalizations may be 1.8 times higher than the reported number. If these estimates are accurate, the true rate of hospitalization risk for the unvaccinated population is 1.6% and as high as 0.2% for the vaccinated population. In either case, the public's misunderstanding of risk is roughly just as inaccurate. One criticism of these inflated estimates is that they assume that many people were hospitalized while asymptomatically carrying the SARS-CoV-2 virus, leading to an undetected case. The problem with this reasoning is that it would count people admitted to the hospital for non-COVID reasons who coincidentally had an asymptomatic infection. These cases were correctly omitted from official statistics since the absence of symptoms cannot cause hospitalization.

A more serious limitation is that we count each admission from COVID-19 into a hospital as a unique person. In fact, we know from scholarly research that some patients are readmitted multiple times. One paper estimates that 9% of COVID-19 patients were readmitted to the hospital. This implies that, at minimum, our hospitalization estimates should be multiplied by 0.91 to capture only hospitalizations of unique individuals. Doing so would shrink both hospitalization risk estimates, and they would still both be well below 1%.