Did you know that two U.S. Presidents, James K. Polk and Warren G. Harding, share the same birthday, November 2nd? With 365 days in a year, it might seem like a rare coincidence for two out of 44 Presidents to have the same birthday. However, in a group of just 44 people, there’s actually a 93% chance that two of them will share a birthday. Let’s explore why this happens.
To make this easier to understand, let’s think about how many people you need in a room to have a better than 50/50 chance of two people sharing a birthday. Imagine you’re the first person in the room. There’s a 100% chance that your birthday is unique to you. When a second person enters, they have 365 possible birthdays, and 364 of those won’t match yours. So, there’s a small chance of a shared birthday.
When a third person joins, there are 363 possible birthdays left that don’t match any already in the room. To find the chance that all three people have different birthdays, we multiply the probabilities of each person having a unique birthday. With each new person, the chance of a shared birthday increases.
Surprisingly, you only need 23 people in a room to have a greater than 50% chance that two of them share a birthday. This is known as the “birthday paradox.” It seems strange because our brains aren’t great at understanding how probabilities work, especially when they grow exponentially.
To illustrate exponential growth, imagine folding a piece of paper in half. Each fold doubles the thickness. If you could fold it 42 times, the paper would reach the moon! This shows how quickly things can grow when they double repeatedly.
Now, let’s make it personal. How many people would you need in a room to have a 90% chance that one of them shares your specific birthday? Each new person added has the same chance of not sharing your birthday. To reach a 90% chance, you’d need 840 people.
Have you ever noticed that you seem to have a friend with a birthday almost every day on social media? To have a friend for every day of the year, you’d need 365 friends. However, the chances of perfectly filling each day are slim. As you add more friends, you’ll quickly fill many birthdays, but the last few will take much longer.
On average, you’d need 2,153 friends to cover every birthday. To have a 90% chance of hitting every single day, you’d need over 21,000 friends. So, if you’re aiming to wish “happy birthday” every day, you might need to start adding more friends!
Gather your classmates and simulate the birthday paradox. Form groups of 23 and record how often two people share a birthday. Repeat this experiment multiple times to see if the results align with the predicted probability of over 50%. Discuss your findings with the class.
Using graph paper or a digital tool, create a chart that shows the probability of two people sharing a birthday as the number of people in a room increases. Start with 1 person and go up to 50. Analyze the chart to understand how quickly the probability increases.
Take a piece of paper and fold it as many times as you can. Measure the thickness after each fold and record the results. Discuss how this relates to exponential growth and compare it to the concept of the birthday paradox.
Use a calculator or a spreadsheet to calculate the probability of at least two people sharing a birthday in groups of different sizes. Start with 5 people and increase the group size by 5 each time, up to 50 people. Share your calculations with the class and discuss any patterns you notice.
Calculate how many people you would need in a room to have a 90% chance that one of them shares your specific birthday. Use the formula provided in the article and compare your results with your classmates. Discuss why the number is so high compared to the general birthday paradox.
**Sanitized Transcript:**
James K. Polk and Warren G. Harding: Two Presidents both born on November 2nd. There are 365 possible days that a person could be born, so out of just 44 Presidents, it’s an interesting coincidence that two would share the same birthday. Actually, no. In a group of 44 people, there’s a 93% chance that two of them could share a birthday. Here’s why.
First, let’s simplify the problem a bit. Let’s figure out how many people you’d have to get together in a room to have just better than a 50/50 chance of a birthday match. Assuming none of them are twins or triplets, and that you’re not at a convention for people whose birthday is on a specific date. You’re the first person in the room. There’s a 100% chance that your birthday is your birthday. What are the chances that the second person to walk in the room doesn’t share your birthday? Well, they have 365 birthdays to choose from, and 364 won’t match yours, so there’s a very small chance of a birthday in common.
When the third person walks in, there are 363 possible birthdays left that don’t match any birthdays already in the room. But how do we put these together? When we combine the odds of independent choices together, we multiply their probabilities.
So the chance that you, person number 2, and person number 3 have unique birthdays, where no one shares, is about 99.2%. With every new person we add to the room, there’s one fewer birthday available, and we continue to multiply the combinations.
23 people. That’s all the people we need before you have a greater than 50% chance of two sharing a birthday. This is known as the birthday paradox. It goes against our intuitions because our brains are not great at figuring the power of chance. Sure, in that room, there’s only 22 possible combinations of your birthday with someone else’s, but there are 253 combinations of everyone’s birthdays. Our brains have trouble imagining these combinations, especially when estimating things that grow exponentially.
For example, how many times do you think you’d have to fold a piece of paper in half before its stacked height reached the moon? Well, we fold it once, two sheets, fold it again, and we have four where we once had one. If I could keep folding this piece of paper indefinitely, by 41 folds we’d reach over halfway to the moon; we only need one more fold to cover all that remaining distance. 42 folds. That number does seem to come up a lot.
Now you understand how few people it takes to get one common birthday, but let’s make it personal. What size group would we need to get, say, a 90% chance of one of them sharing your specific birthday?
In this case, every new person that we add to the room has the same chance of not sharing your birthday. The chances of not sharing combine with every new person we add to the room.
To have a 90% chance of one person having your birthday, you’d need 840 people. I’ve been noticing on social media lately that I seem to have at least one friend with a birthday every single day. But on closer examination, it’s not actually every single day. I’ve got most birthdays filled, but a few are still open.
So how many would I have to have to be able to type “happy birthday” on a friend’s wall every single day? Hypothetically, you’d only need just 365: one for every day, but as we’ve seen, the chances of that happening are pretty small.
The first person you add will have a unique birthday. The second person probably checks another birthday off the list, though there’s a tiny chance they share the first person’s birthday. The third person could be a new birthday or be the same as birthday one or two.
As we go on and on, adding new friends, we will have fewer open birthday slots to fill, and many of our birthdays will start to have multiple people per day. We check off a large number of birthdays very quickly, but the last few will take way longer than we’d expect.
If everyone on social media added friends until they filled every birthday, the average number needed would be 2,153. Of course, any individual person may need way more than that. To have a 90% chance of hitting every birthday, you’d have to add more than 21,000 friends. So, get to clicking!
Birthday – The day of the year on which a person was born, often used in probability to discuss the likelihood of shared birthdays in a group. – In a group of 23 people, the probability that at least two people share the same birthday is surprisingly high.
Paradox – A statement or situation that seems contradictory or impossible but may be true, often used in probability to describe unexpected outcomes. – The birthday paradox illustrates how our intuition about probability can be misleading.
Probability – The measure of the likelihood that an event will occur, expressed as a number between 0 and 1. – The probability of rolling a six on a fair die is 1/6.
Chance – The likelihood of a particular outcome occurring, similar to probability but often used in a more informal context. – There is a 50% chance of flipping a coin and getting heads.
People – Individuals or members of a group, often used in probability problems to calculate outcomes involving human participants. – If 30 people are in a room, what is the probability that two of them have the same birthday?
Unique – Being the only one of its kind, often used in probability to describe outcomes that do not repeat. – Each person has a unique set of fingerprints, making it highly improbable for two people to have the same ones.
Exponential – Relating to a mathematical function involving an exponent, often used to describe rapid growth or decay. – The exponential growth of bacteria can be modeled using the function N(t) = N0 * e^(rt).
Growth – An increase in size, number, or amount, often modeled using mathematical functions in probability and statistics. – The growth of a population can be predicted using exponential functions in probability models.
Friends – People with whom one has a bond of mutual affection, often used in probability problems involving social networks or groups. – If you have 10 friends, what is the probability that at least two of them share a birthday?
Share – To have or use something in common with others, often used in probability to describe events that overlap. – The probability that two people in a group of 23 share a birthday is higher than most would expect.
