Check your intuition: The birthday problem - David Knuffke

Summary

The birthday paradox reveals a surprising truth: in a group of just 23 people, there’s over a 50% chance that two will share a birthday. Using combinatorics, we calculate the probability of no match by considering the odds of each person having a unique birthday. The surprisingly high odds of a match come from the large number of potential pairs in the group, which grows quadratically with group size. This counterintuitive result shows how math often challenges our intuition, and that coincidences, like winning the lottery twice, may not be as improbable as they seem.

Takeaways

• 😀 A group of 23 people has a 50.73% chance of two people sharing the same birthday, despite there being 365 days in a year.
• 😀 The birthday paradox seems counterintuitive because our brains struggle with grasping large combinations and probabilities.
• 😀 To calculate the probability of a birthday match, it's easier to first calculate the probability that no one shares a birthday.
• 😀 The odds that two people have different birthdays are 364 out of 365, or 99.7%.
• 😀 As more people are added to the group, the number of possible pairs increases rapidly, leading to higher chances of a match.
• 😀 In a group of 23 people, there are 253 possible pairs, which is why the probability of a match exceeds 50%.
• 😀 The number of possible pairs in a group grows quadratically, meaning that adding more people increases the probability exponentially.
• 😀 In a group of 70 people, there are 2,415 pairs, and the probability of at least two people sharing a birthday exceeds 99.9%.
• 😀 Our intuition often fails when dealing with non-linear functions like the quadratic growth of possible pairs in a group.
• 😀 The birthday problem highlights how math can show that events that seem unlikely, like coincidences or improbable outcomes, are actually more common than they appear.
• 😀 The birthday paradox is a reminder that coincidences are not always as coincidental as they seem, and probability can reveal unexpected outcomes.

Q & A

What is the birthday problem?

The birthday problem asks how many people need to be in a group before there is a greater than 50% chance that two people share the same birthday. Surprisingly, the answer is 23 people.

Why does the birthday problem seem counterintuitive?

Our intuition suggests that with 365 possible birthdays, it should take a large group for two people to share the same birthday. However, the probability of a match increases quickly due to the large number of possible pairs as the group size grows.

How is the probability of a birthday match calculated?

To calculate the probability of a birthday match, it's easier to first calculate the probability that no one shares a birthday and subtract that from 100%. This involves multiplying the probabilities that each person has a unique birthday.

What are the odds that no one in a group of 23 people shares a birthday?

In a group of 23 people, the probability that no one shares a birthday is 49.27%. This means there is a 50.73% chance that at least two people will share the same birthday.

Why is the probability of a birthday match higher than expected in smaller groups?

The probability is higher because the number of possible pairs increases rapidly as the group size grows. For 23 people, there are 253 possible pairs, each with a chance of sharing a birthday.

How does the number of possible pairs grow with group size?

The number of possible pairs grows quadratically with the size of the group. For instance, a group of 5 people has 10 possible pairs, while a group of 10 has 45, and a group of 23 has 253 pairs.

Why do our brains struggle with the birthday problem?

Our brains have difficulty grasping non-linear functions, such as the rapid increase in the number of possible pairs in a growing group, which is why the result seems counterintuitive.

What happens to the probability of a birthday match in a group of 70 people?

In a group of 70 people, there are 2,415 possible pairs, and the probability that two people will share the same birthday is greater than 99.9%.

What role does combinatorics play in solving the birthday problem?

Combinatorics helps calculate the probability by focusing on the number of possible combinations of pairs and determining the likelihood that none of them share the same birthday.

What does the birthday problem teach us about probability and coincidences?

The birthday problem demonstrates that things that seem unlikely, like shared birthdays in small groups, are not as improbable as they appear. It shows how mathematical probability can make certain coincidences more predictable than expected.