The Lancet reports true breakthrough infection rates - CDC has no good data.

British studies show breakthrough infection rates are extremely low, details follow:

The CDC doesn’t record the number of breakthrough infections, ONLY the ones which require hospitalizations, therefore the British study is definitive.

More than one million people who had one or two vaccination shots 

The mortality rate reported in the UK for those who are hospitalized with COVID-19 (any strain but mostly Delta in the UK) is nearly one-third.

Breakthrough infections are those which occur in people who have had at least one vaccination shot.

The UK measured the infections by self-reporting and testing so they include ALL people who catch COVID-19 EVEN IF THEY HAVE NO SYMPTOMS.

The results are great news because only 0.5% of those who had one of the two shots in the multi-shot vaccines developed COVID.

For those who were fully vaccinated the numbers were half as many, only 0.2%.

The following is a link to the original Lanced paper, it is extremely detailed and even more difficult to understand.

Risk factors and disease profile of post-vaccination SARS-CoV-2 infection in UK users of the COVID Symptom Study app: a prospective, community-based, nested, case-control study - The Lancet Infectious Diseases