When the evidence makes you almost certain something is true, use must. It's a strong guess, not a fact you've checked. Careful: you already know must for rules (You must wear a helmet). Same word, two jobs — and the thing that tells them apart is why you're saying it: a guess reads evidence and works out what is probably true, while a rule tells someone what to do. If you can put so in front of the evidence (The lights are on, so they must be home), it's a guess.
must: you're almost certain
The lights are on, so they must be home. (a guess from evidence)
She's yawning. She must be tired. (a guess from evidence)
Not a guess: You must show your ticket. (a rule — nobody is guessing anything)
Try it: Amina hears loud music and voices through Diego's wall, and his shoes are by the door. Which fits how sure she is? (He must be home. / He might be home.) Answer: He must be home. — The music, the voices, and the shoes make her almost certain, so use must: He must be home. "Might" would mean she's only half guessing, but the evidence here is strong.
can't: you're sure it's NOT so
When you're almost certain something is NOT true, use can't. (Not "mustn't", that means forbidden, which is a different idea.)
His car is still here, so he can't be at the office.
She just ate. She can't be hungry.
That can't be right, the total is way too high.
Not "She just woke up. She mustn't be tired." — say "She just woke up. She can't be tired.". For "I'm sure it's NOT so", use can't, not "mustn't".
Try it: Lars sees that Mei's coat, bag, and keys are all still hanging by the door. Which fits how sure he is she's still here? (She can't be out. / She might be out.) Answer: She can't be out. — All her things are still here, so he's almost sure she hasn't gone out: She can't be out. "Might" would mean he's unsure, but the evidence makes him certain it's not so.
might: it's possible, you're not sure
When something is possible but you're not sure, use might (also could or may). It sits in the middle of a scale of certainty: must = you're sure it's true · might = maybe, it's just possible · can't = you're sure it's not true. All three take the base verb (might be, might know, might rain).
I'm not sure where Sam is. He might be at work.
Take an umbrella. It might rain later.
She isn't answering. She might be asleep, or she might be busy.
Not "He might to be at home." — say "He might be at home.". After might, use the base verb with no "to": might be, not "might to be".
Try it: There's strong evidence: smoke is pouring from the kitchen. There's only a weak hint: the kitchen is a little quiet, but that could mean anything. Match each one. Which fits each? Answer: There's strong evidence: smoke is pouring from the kitchen. → 1 → Something must be burning. · 2 → Someone might be cooking. There's only a weak hint: the kitchen is a little quiet, but that could mean anything. → 1 → Something might be burning. · 2 → Someone must be cooking. — Strong evidence (smoke) makes you almost certain, so must: Something must be burning. A weak hint leaves you only half guessing, so might: Someone might be cooking. Strong evidence takes must; a weak hint takes might.