You seem to be mixing terms up here.
Your post title is "probability of a type I error", but the question you have quoted says "probability of a type II error". You then go on to describe a type 1 error.
For avoidance of ambiguity (source: wikipedia):
Type 1 Error: The probability of rejecting H0 when it is in fact true, ie the chance of being in the rejection region when H0 is true...by definition this is equal to
Type 2 Error: The probability of retaining H0 when it is in fact false, ie the chance of being in the acceptance region when H0 is false...i dont know any way to deduce this from the information you posted in the question.