# Dating problem probability

Having 2 matches is possible, but only happens about 18% of the time. Another interesting observation is that if a match occurs, it is mostly likely that there is only one match (such as the example discussed here).There are many colorful descriptions of the matching problem. One example is that of married couples going to a ball room dancing class.The probability of having at least one match is roughly 0.63.

A previous blog post (Tis the Season for Gift Exchange) presents an example involving gift exchange.

Each person attending a party brings a gift for gift exchange.

The derivation of this probability is based on the inclusion-exclusion principle and is discussed in the blog post called The Matching Problem.

Even though the probability of having at least one match is a function of (the number of items), the probability converges to pretty rapidly.

Further suppose that the secretary stuffs the letters randomly into envelops.

What is the probability that every letter is matched correctly, or that no letter is matched correctly or that exactly letters are stuffed into the correct envelops? What is the probability that at least one letter is stuffed into a correct envelop?It turned out that there was a student who had the same position in both orders.Such a student is called a match (see the following table).In another blog post (A lazy professor who lets students do their own grading), a professor randomly returns quizzes to the students for grading.A match occurs if a students is assigned his or her own quiz.I recently gave an exam in my statistics course, which turned out to be an excellent example for the matching problem, a classic problem in probability. I wrote down the names of the students in the order of turning in the exam.

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