Who throws the dice for you?

[Angrboda]

Premise 4. If our reality is dependent on a supernatural element, then we should observe events which cannot be explained scientifically.
Premise 5. We observe events for which it is virtually impossible to explain scientifically.
Conclusion: Therefore our reality is dependent on a supernatural element.

Where is the fallacy in this argument?

This is a classic example of affirming the consequent, therefore the conclusion doesn't follow. And the reason you've retreated to supernatural causes is because supernatural is defined as the opposite of 'explained scientifically'; if it isn't shown to have a natural, scientific cause, then it's supernatural. You've picked terms which turn your "argument" into a trivial tautology based on the definition of the terms. (You've chosen an excluded middle which is true by definition. Supernatural in this case simply means "not yet scientifically explained," and so it can be substituted for supernatural in your conclusion, yielding the conclusion that our reality is dependent on things that have not yet been scientifically explained. Possibly so, and possibly not. Either way, it's hardly a stunning result.)

Affirming the consequent, sometimes called converse error or fallacy of the converse, is a formal fallacy of inferring the converse from the original statement. The corresponding argument has the general form:

• If P, then Q.
  Q.
  Therefore, P.

An argument of this form is invalid, i.e., the conclusion can be false even when statements 1 and 2 are true. Since P was never asserted as the only sufficient condition for Q, other factors could account for Q (while P was false).