Imagine a group of people. How big do you think the group would have to be before there's more than a 50% chance that two people in the group have the same birthday? Assume there are no twins, every birthday is equally likely, and ignore leap years.
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I love this problem because it shows that things that seem impossible, like having the same birthday as a classmate or family member, are actually possible. Sometimes coincidences aren't as coincidental as they seem. This is a great problem to give to students as a challenge during a probability unit. A similar problem that I also enjoy would be the lottery problem. These problems help students connect mathematics to the real-world and maybe even their own lives (if they happen to share a birthday with a family member or win the lottery).