The exposition is the introduction of background information about the characters, the setting, and anything else we might need to know in order to better understand the story. Often, exposition comes at the beginning of a story, but some stories just drop readers into the middle of the action and lack exposition altogether.
It can be argued that the whole of part 1 of "The Bet" constitutes exposition. In this section, readers meet the "old banker" and learn of the party that took place fifteen years prior to the start of the present action. The narrator introduces the topic of discussion at this party, how the banker and one of his guests, a lawyer, disagreed about the preferability of capital punishment and life imprisonment. The lawyer suggests that both punishments are immoral but that imprisonment for life is still somehow preferable to the death sentence. As a result, the banker bets the lawyer two million rubles that the lawyer would not last five years in solitary confinement, and the lawyer boasts that he could last not only five years but fifteen. And so the bet is made.
In part 1, we also learn how the lawyer has spent his time in imprisonment: what he has read, his hobbies, his requests. All of this information helps to inform the reader of the background of the story so that the banker's fears and the lawyer's ultimate decision can be better understood.