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Can 1 mole of H2 react partially with one mole of O2?

I am currently studying stoichiometry (high school level) and we came across the following equation in school:

2 H2 + 1 O2 = 2 H2O

which is clear to me. We learned in school that reactions in chemistry only happen if there are enough atoms/molecules to react (so every molecule/atom can react), therefore the following reaction won't do anything:

1 H2 + 1 O2 = nothing happens

So now my first question: Why does the following not happen:

1 H2 + 1 O2 = 1 H2O + 1 O

And, as a follow up question, why does the following not work?

1 mole of H2 and 1 mole of O2 won't make 1 mole of H2O and 1/2 mole O2.

What do I do wrong? (maybe I simply misunderstood our teacher)

Are there any explanations for this? I'm especially interested in the answer of my example with 1 mole of H2 and 1 mole of O2. According to our teacher nothing will happen at all. I would assume that at least all H2 molecules react with half of the oxygen molecules to H2O - leaving half a mole of O2 left over. (otherwise this would mean that all molecules "know" each other so they are "aware" that not all molecules/atoms can react)

Can somebody help me? Or point out what I do wrong?

Many thanks

stupidSheep