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All I need to make a comedy is a park, a policeman and a pretty girl.
―Charlie Chaplin
The number three holds a lot of significance in many cultures, literary works, as well as in science. If we look at popular beliefs, the Japanese, associate this number, with truth, courage, and compassion― the 3 Treasures that every human being should possess. In literature, the number three is used to stress the importance of a particular event or object, or point of view; for example, in Charles Dickens' book, A Christmas Carol, the main character Ebenezer Scrooge gets 3 visits from 3 different ghosts. The third ghost was the one that scared him the most, and it showed him the importance of Christmas. Do you remember the first thing we were taught in science about matter? The three states of matter are solid, liquid, and gaseous. The number 3 is quite significant in science, literature, arts, and comedy. The Rule of Three has been vastly applied in comedy, and has found great success.

The Rule of Three in comedy is actually a rule that is followed, when a comedian is either writing something new, or when he is presenting an old joke in front of an audience. Making people laugh is considered as one of the most difficult things in the world. It takes skill, a straight face, quick wit, and an amazing sense of timing. In the world of comedy, timing is the key to everything. This Buzzle article explains the usage of the Rule of Three or the comic triple.
The Explanation
The Rule of Three is also known as trebling, and usually when it applies to comedy, it is in relation to 3 stock characters, or instances. Where the first two have a small sense of normalcy in them, the third is completely stereotyped, as a result of which the outcome is absolutely hilarious. The comic triple multiplies the effect of humor in a speech. Let's analyze a joke:

Three men enter a bar, one is an American, one is a Russian, and the third is an Indian. Now, all three are upstanding citizens, who have never got even a parking ticket. Suddenly, a fight erupts in the bar, and all three die. Since they have never sinned in their life, all three go to heaven, where they meet an angel who grants them three wishes. The American thinks for a moment, and tells the angel that he wishes to drive a Ferrari from the east coast of USA to the west coast. The angel grants his wish in an instant. The Russian says that he has never ridden a Harley Davidson, and his wish is granted too. It's the Indian's turn now, he ponders about this, and tells the angel that he has never had a cigarette in his life. The angel sends him to a room that is filled with the best cigarettes in the world. A little while later, the American returns all happy and thanks the angel, so does the Russian. The Indian, on the other hand, returns all grumpy and irritated. When asked why, the Indian says, "you put me in a room full of cigarettes, but you didn't give me a lighter!"
Analysis
► 3 stock characters- American, Russian, and Indian
► 3 wishes- driving a Ferrari, riding a Harley Davidson, and smoking a cigarette

What was different with the last scenario, was that only the Indian was dissatisfied even though his wish was granted. And why was he unhappy? It was because the angel had not given him the lighter, which is the most obvious thing you need to have a smoke. In the above joke, the same thing (driving and riding, essentially the same) is repeated twice, but the last scenario has a twist to it. If the Indian's wish was granted to him with a lighter as well, then the joke would not have been funny at all.

Therefore, the Rule of Three, means, having three elements in a joke, out of which the first two elements are completely normal, and they have normal responses and reactions. The third, however, is the one that has something out of the blue happening to him, which makes the joke funny. So, the first two elements and situations are repeated twice in two different ways, but the third person, and his scenario is absolutely different, as compared to the first two. The Rule of Three is applied to add humor in a story, but to only the last element, by creating a buildup to it. This rule is a classic joke structure, wherein, you have a setup, the buildup to the end, and the punchline and the end.
Let's look at a joke, that involves one person, but spans over three different instances
It was a cold day of winter, and little Betty was walking home from school. She comes across this small patch on the sidewalk that is frozen, and she has to cross it. So, she looks at it and carefully proceeds, due to which she does not fall. The second day, she crosses it with caution, but not like the previous day, where she took baby steps to do so. She is thrilled about not slipping and falling, and is very confident about this icy patch. Now, the third day dawns, and she is skipping back home, with no worry in the world. She comes across the iced sidewalk and continues to cross it jauntily. Just when she reaches the middle, her foot slips, and she falls flat on her face!

In the above joke, if she had carefully crossed the pavement the third time too, she would not have slipped and fallen, and therefore, it would have just been a normal story about some little girl walking back home. It would not have been interesting, neither would it have been funny. The Rule of Three adds that element of interest to a story and converts it to a joke.
Another example of the rule of 3 used in a joke
Mr. Dibble had an old Fiat. He was driving on the freeway in Germany when his car broke down. He tried to see what was wrong, but he could not figure out the problem. A little while later, a gentleman in a brand new Porsche Carrera GT sees him stranded, and offers to tow his car. So, Mr. Dibble, all happy, sits in his car while it is getting towed. Suddenly, a blur of blue passes by, and the Carrera driver realizes that he just got overtaken by a BMW 6 Series Convertible. His ego cannot take the fact that another car overtook him, and he starts to chase, completely forgetting that he has a Fiat to tow. A small distance away, a patrol officer sees this and calls his superior and tells him, "Sir, I just saw a BMW zoom past me with a Porsche Carrera GT hot on its heels. What I don't understand, is how an old Fiat is keeping pace with the Porsche, 'cause he is continuously honking, so that the Porsche gives him way!"
Rule of Three in Quotes and Sayings
We have jokes, stories, and movies that are funny, but we also have quotes that are absolutely hilarious. We have quotes from notable personalities like Groucho Marx, Melvin Helitzer, Jerry Seinfeld, Laura Kightlinger, Jon Stewart, etc., whose quotes are absolutely hilarious, but still reflect on reality. The Rule of Three is evident in the following quotes.
"I can't think of anything worse after a night of drinking than waking up next to someone and not being able to remember their name, or how you met, or why they're dead."
- Laura Kightlinger
"I celebrated Thanksgiving in an old-fashioned way. I invited everyone in my neighborhood to my house, we had an enormous feast, and then I killed them and took their land."
- Jon Stewart
"One tequila, two tequila, three tequila, floor."
- George Carlin
"There are only three things women need in life: food, water, and compliments."
- Chris Rock
"I, not events, have the power to make me happy or unhappy today. I can choose which it shall be. Yesterday is dead, tomorrow hasn't arrived yet. I have just one day, today, and I'm going to be happy in it."
- Groucho Marx
"Tragedy is when I cut my finger. Comedy is when you fall into an open sewer and die."
- Mel Brooks
"Men are liars. We'll lie about lying if we have to. I'm an algebra liar. I figure two good lies make a positive."
- Tim Allen

In comedy, timing is everything. Making someone laugh is a skill that everyone wants to possess, but very few can pull off. Using the Rule of Three is the most basic technique applied in comic writing and stand-up comedy. It's the easiest rule to abide by and the easiest to learn very quickly. All you need is an audience, quick wit, and impeccable timing.