October 5th is the Most Popular Birthday! The Birthday Paradox Explained
In honor of the estimated 968,000 Americans blowing out their candles in early October, the Birthday Paradox is an interesting learning exercise to help math students grasp the probabilities of a pair.
Beware. Some of these statistics are well beyond the elementary “pumpkin math” exercises in this month’s DreamBox Learning math activities calendar, but the little ones should still be able to walk away with the general idea of how the birthday paradox works.
The birthday paradox asks the question, “what’s the probability that, in a set of randomly chosen people, will any given pair share the same birthday?” We call it a “paradox” because people have a hard time assuming someone else in the general vicinity shares the same birthday. We’re just too self centered.
In addition, our minds have a hard time computing the power of exponents (especially if our little test subjects don’t know what exponents are).
The Birthday Paradox Equation
Let’s walk through this together. First of all, we need to consider what what we’re solving for in this mess of number and letters. Our original question is “What’s the probability a pair of people will share a birthday in a group?”
To start, we need to know the size of the group. Any classroom size should work, but for this example we’re using a group size of 23. Now, we can ask “what’s the probability a pair of people will share a birthday in a group of 23 people?”
Here comes that word “paradox” again. It’s logical to assume 23 people is too small of a group to think two individuals will share the same birthday out of 365 possible days to be born. The math shows otherwise.
If you’re presenting this problem to a group that is a little green when it come to statistics, introduce the coin flipping experiment. When we flip a coin, we know it’s a 50/50 chance that a coin flip will land on tails, but how likely will a coin land on tails 10 times in a row?
You might think to divide the 50% likelihood with the 10 coin flips (.5/10), but you’d be wrong. Instead, you need to multiply the 50% likelihood to the 10th exponent (.5^10) to equal .001 or 1/1000.
It’s a one in a thousand chance that a coin flip will land on tails ten times in a row.
Asking the question of how likely it is for two people to share a birthday in a group of 23 is just like asking, “How likely will a coin land on tails at least once in 23 coin flips?”
You’ll make your head hurt with all the possibilities. Tails can land on the 2nd throw. Maybe on the last throw. Maybe tails hits on every odd number. You can guess it will happen, but you don’t know when.
How do we solve the coin flip problem?
You can’t think about how likely the coin will land on tails without considering how unlikely the coin will land on tails. If it’s less than a .1% chance that a coin will land on heads ten times in a row, much less 23 times in a row, then there’s a 99.9% chance a coin will land on tails at least once in 23 tries.
The trick is to subtract the chance of getting all heads from 1, and we can find the prospect of our desired outcome.
Solving the Birthday Paradox
Good, you’re still with me. Back to the problem of birthdays. Just like the coin flip, let’s ask how unlikely is it for a pair to share a birthday in a group of 23 people?
For the birthday paradox experiment, we also assume the following:
- There are 365 days in a year (forget about those pesky leap years)
- There are no twins in the group of 23
Start off by calculating how many pairs 23 people make. Since any one person out of 23 can make 22 different pairs, multiply 22 with the total number of people (23) and divide by the amount of people it takes to make a pair (2). This equals 253.
Now, you have to ask the question “how likely is it for a pair of people to not share a birthday in a year?” If a person’s birthday is Oct. 5th, that means their birthday occupies 1/365 of the year. We can assume someone else can be born on any of the 364 days of a 365 day year, and not share the same birthday.
Going back to the idea of coin flips, trying to find 253 different birthdays is like 253 tails flips in a row (assuming all birthdays are independent). Use the 253 as the exponent to the total non-matching birthdays (364) over the total days of the year (365):
The chance of a birthday match out of 23 people is:
1 – 49.95% = 50.05%.
It’s over 50%! Which means a group of 23 is more likely than not for two people to share a birthday.
Probability of a Pair Graph
This graph indicates the exponential growth of a probability of a pair based on the size of a group. At 57 people, the probability of a birthday match is 99%, and to reach 100%, you’ll need a group of 366 people.
Now Let’s Play!
Make this into a fun classroom activity by asking a class how likely it is that they share a birthday with someone else in the class. You can also change the experiment. Have the children secretly write down a number between 1 and 365, and ask them how likely is it that someone else picked the same number.
If you’d like a further explanation of the birthday paradox, BetterExplained has step-by-step instructions, as well as a birthday paradox calculator. And if you have an October 5th birthday, comment below!
If you had fun with this October activity, check out our Pumpkin Math Activities Calendar.