“Had to” vs. “must have”

The modal verb must has two past tense forms: had to and must have. Which form we use depends on whether we want to express obligation or if we want to say how certain we are about the probability of something happening. 

This table below shows us the past tense of must and have to and when to use them.

Must, have to
present past
When expressing obligation, we say:
I must go.
or
I have to go.
When expressing obligation, the past of must and have to is always had to:
I had to wash my car yesterday .
We had to be go to bed at 8 o’clock when we were kids.
When expressing a personal opinion about probability (deduction), we mostly use must to express that we feel something is true:

He must be fit if he can run 10 kilometres.
It must be great to be rich.
When expressing a personal opinion in the past, we mostly use must have, NOT had to:

He must have been fit.
He had to be fit.

It must have been great.
It had to be great.