The Disadvantages of Being Absent-minded

Here is an interesting lateral thinking puzzle given in Raymond Smullyan's The Lady or the Tiger?: and Other Logic Puzzles, page 5. Smullyan says in the book that it is a true story that happened to him once. Note that I have modified the quotation slightly as it is written in the first person in the book.

It is well known that in any group of at least 23 people, the odds are greater than 50 percent that at least two of them will have the same birthday. Professor Smullyan was once teaching an undergraduate mathematics class at Princeton, discussing elementary probability theory. He explained to the class that with 30 people instead of 23, the odds would become enormously high that at least two of them had the same birthday.

"Now," the professor continued, "since there are only nineteen students in this class, the odds are much less than fifty percent that any two of you have the same birthday."

At this point one of the students raised his hand and said, "I'll bet you that at least two of us here have the same birthday."

"It wouldn't be right for me to take the bet," the professor replied, "because the probabilities would be highly in my favor."

"I don't care," said the student, "I'll bet you anyhow!"

"All right," the professor said, thinking to teach the student a good lesson. He then proceeded to call on the students one by one to announce their birthdays until, about halfway through, both the professor and the class burst out laughing at the professor's stupidity.

The boy who had so confidently made the bet did not know the birthday of anyone present except his own. Can you guess why he was so confident?

The answer is on the answers page.

Sources used (see bibliography page for titles corresponding to numbers): 31.