20% of the population is unvaccinated, 29% of total hospitalizations are unvaccinated. how much more likely is someone unvaccinated to end up in the hospital

Question

20% of population unvaccinated

29% of patients in hospital are unvaccinated

how much more likely is someone unvaccinated to end up in the hospital

Answer 1

20% of the population are unvaccinated

29% of patients in the hospital are unvaccinated

how much more likely is an unvaccinated person to end up in the hospital?

There must be more information in this problem, such as how large is the population and how many are in the hospital. What percent of the population was in the hospital. Also, there is no necessary causation, especially if the reason someone is in the hospital may be totally unrelated to whether they were vaxed. You could be in the hospital for a car accident or gunshot wound.

If there were a population of 1,000 and 10% in the hospital, then there would be 100 in the hospital with 29 not vaccinated, and 200 of the population not vaccinated. 800 vaccinated and 71 in the hospital who were vaccinated.

Odds of being in the hospital if vaccinated is 29/200 = 14.5%. Odds of being in the hospital if vaccinated is 71/800 = 8.87% You're then 14.5-8.87 = 5.63% more likely to be in the hospital if you're not vaccinated.

But if no one is in the hospital, or it's near zero % of the population in the hospital, there's zero difference or nearly zero difference of your likelihood to be in the hospital whether vaccinated or not.

If 20% of the population is in the hospital, then 58 of 200 unvaxed are in the hospital, which is 58/200 = 29% chance of an unvaxed to be in the hospital. And 2(71)/200 vaxed are in the hospital which = 71% chance of being a hospital patient if unvaxed. 72-29 = 43% greater probability of being a hospital patient if unvaxed.

if 40% of the population is in the hospital, then .29(400) = 116 of the 200 unvaxed are in the hospital. 116/200 = 58% chance of an unvaxed to be hospitalized. And .71(400) = 284 of 800 vaxed are in the hospital. 284/800 = 35.5% chance for a vaxed to be in the hospital. 58-35.5 = 22.5% greater chance to be in the hospital if unvaxed.

if 60% of the population is in the hospital, then .29(600) = 174 of 200 unvaxed are hospitalized which is 174/200 = 87% chance for an unvaxed to be hospitalized. .71(600) = 426 of 800 vaxed are hospitalized, which is 426/800 = 53.25% chance for a vaxed to be hospitalized. 87- 53.25 = 33.75% more likely for an unvaxed to be in the hospital.

if 69% of the population is in the hospital, then .29(690) = 200 of 200 unvaxed are in the hospital, a 100% chance for an unvaxed to be hospitalized. .71(690) = 490 of 800 vaxed are in the hospital, which is 490/800 = 61.25% probability. 100-61.25 = 38.75% better odds an unvaxed is in the hospital than a vaxed.

the odds are anywhere between zero % to nearly 40% more likely you'll be in the hospital if unvaxed.

if more than 69% of the population is in the hospital, that's inconsistent with the given information of 20% and 29% for the unvaxed in the population and in the hospital.

Odds are a small percentage of the population is in the hospital at any one time, so the difference in likelihood of hospitalization whether vaxed or not is also small.

if you're counting total hospitalizations over time, you could get over 100% of the population hospitalized as many might be hospitalized more than once in their life.

Answer 2

Looks like you can set up the Bayes Relation for both vaccinated and unvaccinated:

P(U|H)P(H) = P(H|U)P(U) We don't know P(H), but we want P(hospital|unvaccinated)

P(H|U) = P(H) (.29/.2) because P(U) = .2 and P(U|H) = .29

You can do the same thing for vaccinated which leads to P(H|V) = (.71/.8) P(H) because P(V) = .8 P(H|V) = .71

You can take the ratio of P(H|U)/P(H|V) and that is how much more likely an unvaccinated vs a vaccinated individual will go to the hospital.

This is one way to do it.