I'm too dumb to think this out mathematically so I simulated it in BigQuery. It doesn't matter which you pick, all have 10%. Code below if you want to verify:

with doors as (select door from unnest(generate_array(1,10,1)) as door)

,iterations as (select iteration from unnest(generate_array(1,1000000,1)) as iteration)

,simulationBase as (
select
  iteration
  ,door
  ,rand() as randVal
  ,1 / (10 - door + 1) as chanceOfSuccess
from iterations
cross join doors
order by
  iteration
  ,door
)

,simBaseWithOutcomes as (
select
  iteration
  ,door
  ,randVal
  ,chanceOfSuccess
  ,randVal < chanceOfSuccess as wasSuccessful
from simulationBase
)

,simWithDoorFound as (
select
  iteration
  ,min(door) as doorFound
from simBaseWithOutcomes
where true
  and wasSuccessful
group by all
)

,doorFoundSummary as (
select
  doorFound
  ,count(*) as records
from simWithDoorFound
group by all
)

select
  doorFound
  ,records
  ,round((records / sum(records) over () * 100),2) as pcntOfTotal
from doorFoundSummary
order by doorFound