The Birthday Paradox in R: Unlocking Probability Puzzles with Code
Probability often defies human intuition. One of the most famous examples of this is the Birthday Paradox (or birthday problem), which asks a simple question: How many people need to be in a room together for there to be a greater than 50% chance that at least two of them share the exact same birthday?
Most people guess hundreds, given there are 365 days in a year. The staggering mathematical reality? Just 23 people.
Using R, a powerhouse language for statistical computing and data science, we can easily simulate and calculate this phenomenon. Let’s explore how the birthday problem works and how you can model it yourself using built-in R functions and custom simulations.
Understanding the Math: Why 23?
To find the probability of a shared birthday, it is actually much easier to calculate the opposite: the probability that no one shares a birthday.
If you bring people into a room one by one:
- The first person has a $365/365$ chance of having a unique birthday.
- The second person has a $364/365$ chance of not matching the first.
- The third person has a $363/365$ chance, and so on.
For $n$ people, the probability of all unique birthdays is the product:
$$P(text{no matches}) = frac{365}{365} times frac{364}{365} times frac{363}{365} times dots times frac{365 – n + 1}{365}$$
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