This question tests your understanding of modal verbs and perfect infinitives to express logical conclusions about past actions.

Let's analyze the conversation:

• Person A answers the phone and identifies themselves as "Jack Williams".
• Person B says "Sorry" and then the blank, finishing with "the wrong number".

The context shows that the action of dialing the number happened in the recent past, just before the phone rang. Person B is making a logical conclusion or a guess about what they just did.

Step 1: Identifying the Time of the Action

The key action, "dial the wrong number," is completed. It happened before the moment of speaking. Therefore, we need a verb form that refers to a past action.

Both "Might have dialed" and "Must have dialed" use the perfect infinitive ("have" + past participle), which is used to talk about past time.

Step 2: Choosing the Correct Modal Verb

Now we need to choose the correct modal verb to express the speaker's meaning.

• Must have + past participle is used to express a strong certainty or a logical conclusion about a past event. The speaker is almost sure that this is what happened.
• Might have + past participle is used to express a possibility about a past event. It is a guess, but the speaker is not very certain.

In the conversation, Person B says "Sorry." This apology suggests they are fairly certain they made a mistake. They aren't just guessing; they are concluding that they definitely dialed the wrong number. Therefore, "must have dialed" is the most suitable choice to show this certainty.

Final Answer: The correct option is "Must have dialed". The complete sentence is: "Sorry. I must have dialed the wrong number."

Related Topic: Modals of Deduction (Past)

We often use modal verbs with the perfect infinitive to make deductions about the past.

• Must have + past participle: For when we are almost certain something happened.
  Example: She isn't answering her phone. She must have left the office already.
• Might/Could have + past participle: For when we think it's possible something happened.
  Example: I can't find my keys. I might have left them in the car.
• Can't have + past participle: For when we are almost certain something did not happen.
  Example: He can't have stolen the money; he wasn't even in the country.

Related Formulae

While there isn't a mathematical formula, the grammatical structure is consistent:

Modal Verb + Have + Past Participle (Verb³)

This structure is the key to forming past deductions in English.