A good analysis of "The Bet" would include all the aspects of a literary analysis, which requires certain information be included. This information pertains to the literary devices of the text. Literary devices are of two types: literary elements and literary techniques. Literary elements are the devices that are in all literature, whether long or short story fiction. Some literary elements are conflict, mood, point of view.
Literary techniques are options that the individual writer may choose to enhance the story. These optional literary techniques are termed tropes. Some tropes--which are literary techniques in writing that may optionally be employed by the writer--are personification, metaphor, irony, simile. To write a literary analysis, you must include mention of all the elements and the techniques that pertain to the text.
An analysis of "The Bet" by Anton Chekhov must include, in proper essay form, the following literary elements:
Setting: A wealthy banker's estate in November of the year 1870.
Point of View: Third person limited narrator who knows and communicates only the bankers thoughts, emotions and motivations.
Conflict: Man against man; who will win the bet and how will the banker behave toward the younger man.
To continue your analysis, you must answer for the other literary elements of:
Mood, which is the feeling of the story built from all aspects of language, setting, characters, details;
Tone, which is the attitude of the narrator, which may be different from the mood, as in a calm narrator telling a horror story;
Flashback, like the one that occurs in the second paragraph of "The Bet";
Foreshadowing, which is clues or suggestions that indicate what will happen later;
Indirect characterization and direct characterization;
Then you must include discussion of all the literary technique tropes that you can identify. An analysis of "The Bet" might contain a paragraph like the following:
"The premise of "The Bet" is built on a trope called a metaphor. A metaphor compares two things that are not alike but that shed light on the meaning of the one. For instance, it is a metaphor to say, "Matilda is a brick," meaning she is sturdy and dependable. In paragraph two, the banker says, "lifelong imprisonment kills a man slowly." This is a metaphor because imprisonment does not in fact kill, it imprisons. The metaphor sheds light on the true meaning of imprisonment."
Your analysis would go to identify any other techniques Chekhov used in the story. The four classic tropes are metaphor, metonymy, irony, and synecdoche. Metaphor is defined above. Metonymy is the use of a familiar phrase to represent a larger concept, like the White House represents the President of the U.S. and the President's government. In "the Bet," the metonymy "outer world" refers to all human experience.
Irony is when words or situations present something other than what one expects. "The Bet" has irony at the resolution because with all the precautions, in the end, the younger man escapes out the window. Synecdoche is similar to metonymy because a word or phrase is substituted for something else, but in this case, an abbreviation is substituted, such as saying "the U.S." to mean "the U.S. Olympic Ski Team" in "The U.S. won gold medals in cross-country skiing," or using "word" to mean "word of honor."