There are 30 people in a room. You want to predict the number of people in the room who share birthdays with others in the room. How could you begin to set up a simulation for this scenario is given below
Step-by-step explanation:
First, I’m going to walk through a step-by-step of solving it, and I’ll provide a short explanation at the bottom for why this is the case.
To figure this stat, let’s first realize that, excluding twins, each of the 30 people has an equivalent 365 days of the year that could be their birthday. Therefore, the total combination of all the possible probabilities of birthdays for all of the 30 people is 365 * 365 * 365 * … 30 times or, better expressed, 365^30.
An easier way to solve this problem than solving for the probability that any 2 or more of the 30 people share a birthday is to solve for the probability that all of the people have unique birthdays (non-shared) and subtracting that from 100%.
The first person has 365 possible days that could be their birthday without sharing with someone else. Then, the second person has 364 days that could be their birthday without sharing with someone else, because person 1’s birthday is one of those days. This process goes on for all of the 30 people, until the 30th person has 336 possible days that could be their birthday.
A better way to express the total possible combinations of days in which none of the 30 people share a birthday is 365 * 364 * 363 * … until 336 or, better expressed, 365!/335!
To solve for the total probability that, out of the original 365^30 days, there are 365!/335! of them where no one shares a birthday, we simply divide the latter by the former.
(365!/335!)/(365^30)
If you do this immense calculation, you can solve that the probability that none of the 30 people will share a birthday is 29.36837573%. If you subtract this from 100%, you get the probability that a minimum of two people do share a birthday, which is 70.63162427% or roughly 7/10.
While at first glance, the answer may seem obvious as 30/365 or 335/365 or any other quick calculation, you have to realize that this calculation is an example of stacking probability. While the probability that the first two people don’t share a birthday is quite minuscule, this probability stacks, so to speak, and grows exponentially for every additional person whose birthday you must consider. An easier way of comprehending this is recognizing that, for every additional person, you have to calculate the probability that their birthday does not match with any of the others’ birthdays. For numbers going past just a handful, this number does grow quickly, as each person has a certain number of people with whom they can’t share a birthday, and this is true for every one of that certain number of people.
While this may be hard to wrap your head around, it is simple when done by calculation. It’s just an example of how the intuitive part of your brain tries to solve this problem by going for the quick, easy solution that may not always be accurate. You have to force the deep thinking part of your brain to actually analyze the problem for what it truly is and see that it is a complex probability.
