The Idea in Brief
Peter acknowledges in his introduction of the epistle that the recipients of his letter know about the Lord. That is, he addresses them as "to those who received a faith the same as ours." However, Peter does not assume that they all therefore know the Lord. That is, he states to them in the same first chapter: "Therefore, brethren, be all the more diligent to make certain about His calling and choosing you" (2 Pet 1:10). Peter is therefore addressing those for whom in fact there is some uncertainty as to whether or not they know the Lord, since the historical pattern of "believers" going astray in the Hebrew Bible (e.g., Balaam) was through false teaching, irreverence (mocking the Word of God), sexual immorality, and/or greed: thus Peter cites the Proverb, "The dog returns to his vomit and the washed sow returns to wallow in the mire." Irrespective, the Lord is patient with all, not wishing that any perish, which would include all those people who may know much about the Lord, but yet who STILL do not know Him (as is apparent by their life of debauchery and irreverence).
Peter opens his epistle "to those who have received a faith of the same kind" as ours. The Greek words are τοῖς ἰσότιμον ἡμῖν λαχοῦσιν πίστιν. The literal translation would be "to those who have chanced upon the faith equally valued by us." The participle λαχοῦσιν comes from the verb λαγχάνω, which means to receive by lot or divine appointment.
For example, in the Book of Acts, Peter mentioned that Judas Iscariot had "received his lot," or appointment from the Lord (λαγχάνω in Acts 1:17). Notwithstanding that Judas knew much about Jesus and His teachings, Peter states in the context in the first chapter of Acts that Judas did not know the Lord even though Judas had "received his lot," or appointment from the Lord to be one of the twelve disciples.
Accordingly, in this second epistle of Peter, Peter uses the same verb form (λαχοῦσιν participle) to indicate that his readers "have chanced upon the faith equally valued by us." Peter does not accuse anyone of unbelief, but his choice of words (as was the case with Judas Iscariot in Acts) serves to indicate that there is some uncertainty as to whether or not all his readers know the Lord.
2 Peter 1:10 (NASB)
10 Therefore, brethren, be all the more diligent to make certain about His calling and choosing you; for as long as you practice these things, you will never stumble;
From this point forward, Peter then develops in the epistle how "believers" in the past have fallen away through false teaching, irreverence (mocking the Word of God), sexual immorality, and/or greed with specific mention of Balaam (2 Pet 2:15). He summarizes and describes these "believers" with the Proverb, "the dog returns to his vomit and the washed sow returns to wallow in the mire" (2 Pet 2:22).
In conclusion, Peter uses the participle form of the verb λαγχάνω to indicate that his readers have "chanced upon" the faith that we all value equally. That is, by divine appointment we have all received the saving message of faith. However, like Judas Iscariot, who not only received the same message of saving faith, but who also "chanced upon" (received the divine appointment) to being one of the twelve disciples, there was uncertainty. That is, Judas did not know the Lord despite knowing much about Him. Peter therefore admonishes his readers in this Second Epistle, that the Lord is patient that all be saved, which would include those "believers" who have yet to show that they know the Lord through the example of their lives.